Spin-Skyrmion Tube: 3D Topological Spin Texture

• Spin-skyrmion tubes are three-dimensional topological spin textures that extend a skyrmionic cross-section through a magnetic body, exhibiting unique internal modulations.
• They are characterized by non-uniform Bloch/Néel admixture, tube-axis modulations, and localized dynamics that drive current-induced mobility and transport phenomena.
• Stabilization arises from competing exchange, chiral, magnetostatic and dipolar interactions, with advanced imaging techniques revealing protocol-dependent phase transitions.

A spin-skyrmion tube, usually termed a skyrmion tube (SkT), is a three-dimensional topological spin texture in which a skyrmionic cross-section elongates through the thickness or depth of a magnetic body. In experiment and simulation, SkTs are not simply two-dimensional skyrmions extruded into perfectly homogeneous cylinders. Direct real-space observation in FeGe established that skyrmions can exist as tube-like objects extending through a sample, while three-dimensional vector-field reconstruction showed local deviations from homogeneous Bloch character, tube-axis modulation, and surface-induced collapse of the texture (Birch et al., 2019, Wolf et al., 2021).

1. Definition, scope, and conceptual status

SkTs occur across several classes of magnetic systems, including bulk chiral magnets, confined FeGe lamellae and cuboids, cylindrical nanostrips and nanotubes, synthetic antiferromagnetic multilayers, and frustrated magnets. In this literature, the tube is the relevant three-dimensional object even when a cross-section resembles a conventional skyrmion. A common misconception is that device-relevant skyrmions are intrinsically two-dimensional; the direct observation of isolated SkTs in FeGe, including Bloch point terminations, established that the experimentally realized object can be an extended three-dimensional texture rather than an idealized planar soliton (Birch et al., 2019).

The internal structure of the tube depends strongly on materials class. In synthetic antiferromagnetic multilayers, a hybrid chiral skyrmion tube twists along the thickness and transitions from left-handed Néel-type on one surface, through a central Bloch-type configuration, to right-handed Néel-type on the opposite surface; this structure arises from competition between interfacial Dzyaloshinskii-Moriya interactions and long-range dipolar interactions (Dohi et al., 2024). In cubic helimagnets within the conical state, both vertical and horizontal skyrmion tubes have been analyzed: the vertical object is a cylinder-like core with helical modulation, whereas the horizontal object consists of a pair of merons and can interconvert with the vertical orientation through spring-like intermediate states (Leonov et al., 2021).

2. Internal spin texture and topological descriptors

The cross-sectional topological charge of a skyrmionic texture is commonly written as

$$Q = \frac{1}{4\pi} \int \mathbf{m} \cdot \left(\frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y}\right)\, dx\,dy .$$

This quantity remains central for single tubes, bundles of tubes, and fractional realizations (Tang et al., 2021).

Quantitative three-dimensional reconstruction in FeGe showed that the label “Bloch-type” is only approximate in real confined specimens. The reconstructed textures revealed that the in-plane component of the magnetic induction does not remain purely tangential to the skyrmion core; radial components are often observed, especially near surfaces and at circumferential Bloch walls. The same study reported broken axial symmetry, tube axes that are bent and twisted, correlated modulation along the tube axis with a characteristic length of approximately $80\,\mathrm{nm}$, in-plane magnetic flux leaking between adjacent tubes, and indications of Bloch points or related defects in tightly packed regions (Wolf et al., 2021).

Hybrid and twisted tubes generalize this nonuniformity further. In hybrid chiral tubes, chirality changes systematically across the multilayer thickness rather than remaining homogeneous (Dohi et al., 2024). In frustrated magnets, “twisted skyrmion tubes” exhibit a helicoid pattern along the vertical direction and a position-dependent helicity. Their ansatz is written as

$$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$

with $\Phi = Q_v \phi + \varphi(z)$, so that the cross-section interpolates between Neel- and hedgehog-like two-dimensional skyrmions. The same framework introduces a Hopf index

$\mathcal{Q}_H = \frac{Q_v\,\kappa L_z}{2\pi},$

which is proportional to thickness and characterizes the three-dimensional linking structure of the texture (Saji et al., 26 May 2025).

These results collectively exclude the view of the SkT as a rigid, axially uniform cylinder. The internal helicity, local Bloch/Néel admixture, and end-state topology are intrinsic parts of the object rather than perturbative corrections.

3. Stabilization mechanisms, confinement, and thermodynamics

SkT stabilization depends on competing exchange, chiral, magnetostatic, anisotropy, and Zeeman terms, but the balance is strongly geometry-dependent. In multilayers, hybrid chiral tubes arise from competition between interfacial Dzyaloshinskii-Moriya interactions and long-range dipolar interactions (Dohi et al., 2024). In cylindrical nanostructures with chiral interactions, the controlling competition is between the curvature radius $R$ and the skyrmion radius $R_{\rm sk}$. Wide nanotubes with $R > R_{\rm sk}$ support stable skyrmions that preserve their size and circularity, whereas narrow nanotubes with $R < R_{\rm sk}$ drive the skyrmion phase into a stripe phase. Within the same stability regime $R \gtrsim R_{\rm sk}$, zero-field skyrmions are stable on curved nanoelements with free boundaries (Kechrakos et al., 2019).

Confinement and edges are not merely secondary perturbations. In FeGe lamellae under in-plane magnetic field, SkTs were observed preferentially at edges and corners, showing that sample geometry, demagnetising fields, and edge conditions are critical for metastability. Comparative micromagnetic simulations indicated that tubes readily form at edges in confined geometries, plausibly to avoid energetically costly Bloch points or because of favorable edge states (Birch et al., 2019).

Thermal stability in confined chiral magnets is narrow in equilibrium but can be extended metastably by protocol. In FeGe nanostructured cuboids, skyrmion tubes were observed to stabilize only in a narrow field-temperature region near the Curie temperature $80\,\mathrm{nm}$0. At $80\,\mathrm{nm}$1, increasing an in-plane field $80\,\mathrm{nm}$2 induced a first-order transition from vertical helix to horizontal skyrmion tubes, before further field increase drove the system to the ferromagnetic state. Below $80\,\mathrm{nm}$3, tubes could not be formed by simply ramping the field, but field cooling trapped metastable tube configurations; in the interval $80\,\mathrm{nm}$4, no stable or metastable skyrmion tubes were reported (Jailiang et al., 15 Feb 2025).

In frustrated magnets, twisted tubes are stabilized by competing next-nearest-neighbor exchange interactions, the thickness of the magnet, and the applied magnetic field, while dipole-dipole interactions can lift the degeneracy of the helicity mode and favor Bloch-type helicities (Saji et al., 26 May 2025). This suggests that “tube stability” is not a single mechanism but a family of mechanisms specific to dimensionality, confinement, and the dominant microscopic couplings.

4. Imaging and reconstruction of three-dimensional tube states

Real-space access to SkTs has required methods that can either tolerate in-plane fields or reconstruct full vector information. The first real-space observation of confined magnetic skyrmion tubes in FeGe used resonant magnetic x-ray imaging, specifically x-ray holography and cryogenic scanning transmission x-ray microscopy. These methods exploited x-ray magnetic circular dichroism at the Fe $80\,\mathrm{nm}$5 edge and enabled imaging under in-plane fields, a configuration not easily accessible with electron-based approaches (Birch et al., 2019).

Three-dimensional vector reconstruction was achieved by holographic vector field electron tomography. In FeGe needles, this method provided full $80\,\mathrm{nm}$6 maps with sub-10 nanometer spatial resolution by combining off-axis electron holography tilt series around orthogonal axes and using $80\,\mathrm{nm}$7 to recover the third field component (Wolf et al., 2021). The same measurements enabled spatially resolved exchange and Dzyaloshinskii-Moriya energy-density mapping, thereby connecting imaging directly to the local stabilization mechanism.

Lorentz transmission electron microscopy has been central to phase mapping in confined chiral magnets. In FeGe cuboids, Lorentz TEM with an in-plane-field specimen holder was used under three protocols—zero-field warming, field cooling, and field ramp at fixed temperature—to track transitions among helical, conical, tube, and ferromagnetic states (Jailiang et al., 15 Feb 2025). In chiral-interfaced three-dimensional nanowires, x-ray magnetic ptychography with x-ray magnetic circular dichroism resolved vortex, anti-parallel, hybrid, and fractional skyrmion-tube states, with micromagnetic simulations used to generate projected contrasts for direct comparison (Fullerton et al., 2024).

The imaging literature therefore established three distinct but complementary facts: SkTs are directly observable in real space; their internal vector structure is reconstructable in three dimensions; and their phase stability is protocol-dependent in confined geometries.

5. Current-driven dynamics and transport phenomena

Current-driven dynamics reveal that the SkT is a mobile three-dimensional quasiparticle whose response depends on both global topology and internal degrees of freedom. In magnetic nanotubes, micromagnetic simulations showed that current injected along the tube axis produces a helical trajectory: the skyrmion moves axially and tangentially because of the skyrmion Hall effect. The axial velocity is proportional to current density, while the annular speed increases with nanotube thickness. Because the tangential direction is edgeless, stable propagation survives at very large current density, reported as $80\,\mathrm{nm}$8, with speeds up to approximately $80\,\mathrm{nm}$9 in simulation (Wang et al., 2018).

Synthetic antiferromagnetic multilayers alter the same problem by compensating gyrotropic forces across antiferromagnetically coupled layers. Current-induced dynamics of SyAF skyrmion tubes showed that, for a fixed total thickness, increasing the number of ferromagnetic layers makes each sublayer thinner, decreases deformation, increases rigidity, and raises the critical current before destruction. The reported examples were $$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$0 for $$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$1 and $$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$2 for $$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$3. For an even number of equally thick ferromagnetic layers, the skyrmion Hall effect is eliminated, with steady motion described by

$$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$4

Larger damping suppresses deformation but slows motion, whereas larger fieldlike torque increases both deformation and speed (Xia et al., 2021).

Hybrid chiral tubes in SyAF multilayers add a further layer of dynamics: a non-reciprocal skyrmion Hall effect in the flow regime. The degree of magnetic compensation was defined as

$$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$5

Experimentally, the non-reciprocity was negligible in the creep/depinning regime, emerged at high velocities of approximately $$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$6, and was strongest for the lower-compensation sample with $$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$7; it vanished near $$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$8, where the system favored homochiral Néel tubes. Unlike earlier two-dimensional reports, the effect did not arise from velocity asymmetry. Micromagnetic simulations and theory instead tied the phenomenon to asymmetric dynamic oscillation of the tube helicity. The analysis used the spin-orbit-torque efficiency tensor

$$\boldsymbol{m}(\rho, \phi, z) = (\cos\Phi \sin\Theta,\, \sin\Phi \sin\Theta,\, \cos\Theta),$$9

and the dissipation tensor

$\Phi = Q_v \phi + \varphi(z)$0

with minimal shape-distortion contributions inferred from the latter (Dohi et al., 2024).

For twisted skyrmion tubes in frustrated magnets, spin-orbit torque drives helicity dynamics directly. The equilibrium helicity precesses with

$\Phi = Q_v \phi + \varphi(z)$1

and the associated emergent electric field generates

$\Phi = Q_v \phi + \varphi(z)$2

In that framework, the tube is not only a transport carrier but also a source of emergent electromotive response (Saji et al., 26 May 2025).

6. Derived tube states, collective objects, and functional roles

SkTs serve as building blocks for more complex three-dimensional topological objects. In FeGe, a circular spin spiral was shown to bind a bunch of skyrmion tubes into a skyrmion bundle. These bundles were imaged, created, and manipulated with integer topological charges from negative values up to at least $\Phi = Q_v \phi + \varphi(z)$3. For nonzero $\Phi = Q_v \phi + \varphi(z)$4, the bundle behaved as a quasiparticle under electric pulses and exhibited a skyrmion Hall angle of $\Phi = Q_v \phi + \varphi(z)$5; for $\Phi = Q_v \phi + \varphi(z)$6, the bundle propagated collinearly with the current flow without a skyrmion Hall effect (Tang et al., 2021).

Field-driven tube metamorphosis in cubic helimagnets extends this compositional view. Horizontal skyrmions can swirl into spring-like intermediate states and subsequently squeeze into vertical skyrmions with both polarities. The same setting supports target-skyrmions, including a multiple-topological-charge type, formed by minimizing the interaction energy between vertical and horizontal skyrmions within the conical background (Leonov et al., 2021).

Non-integer variants have also been realized. In three-dimensional printed ferromagnetic double-helix cobalt nanowires with adjacent left-handed and right-handed regions, the coexistence of vortex and anti-parallel states at the chirality interface produced fractional Bloch skyrmion tubes at room temperature and zero magnetic field. The topological charge in the fractional region rose continuously from approximately $\Phi = Q_v \phi + \varphi(z)$7 to $\Phi = Q_v \phi + \varphi(z)$8, and the same structures supported reconfigurable zero-field states including pure vortex and mixed skyrmion-vortex states (Fullerton et al., 2024).

SkTs also act as nonplanar magnonic elements. In a magnetic nanoprism, an isolated skyrmion tube induces spatially separate internal and edge channels for spin waves; the internal channel has a narrower energy gap, supports lower-frequency propagation, and is less susceptible to magnetic field than the edge channel. Signal transmission in these waveguides results from coherent propagation of locally driven eigenmodes, specifically the breathing and rotational modes (Xing et al., 2019).

A plausible implication is that the technologically relevant degree of freedom of a SkT is increasingly not just its center-of-mass position, but its internal helicity, twist, bundle charge, or channel structure. In that sense, the modern literature treats the spin-skyrmion tube as a three-dimensional topological object whose static form, dynamics, and functionality are inseparable.