Main

Magnetic skyrmions manifest as spin vortices, whose topological protection drives particle-like properties and condensation into thermodynamically stable phases in an external magnetic field1,2,3,4. Most commonly observed as two-dimensional (2D) structures in metallic materials5,6,7,8,9,10, their quantized emergent magnetic flux1,11,12,13, which characterizes the Berry phase an electron accrues when following the local magnetization adiabatically14, generates unique transport phenomena15,16,17, multiferroic behaviour18 and electric controllability via ultralow current densities12,19. Such a culmination of features sees skyrmions as promising candidates for next-generation low-power spintronic information-processing and storage devices20,21,22.

In three dimensions, the uniform stacking of the 2D skyrmion spin structure produces skyrmion tubes elongated along the external magnetic field direction, thought to penetrate surface to surface23,24. In physical bulk crystal systems at non-zero temperature, a finite density of defects exists, interrupting the skyrmion string propagation25. Because the emergent flux that defines skyrmions is quantized, their nucleation and termination is mediated by emergent magnetic charges that must also be quantized13. Skrymion tube segmentation and branching via emergent magnetic monopoles and antimonopoles (denoted S+, S−, B+ and B−, respectively) are believed to mediate skyrmion topological transitions26. Motion of such defects in response to changes in external parameters, such as field or temperature conditions, have been proposed to drive a change in skyrmion topology through the unwinding of individual skyrmions25,26 and the zipping and unzipping of neighbouring skyrmion tubes13,26,27. Since total emergent charge is conserved, skyrmionic transitions can only take place in three dimensions when emergent (anti)monopoles are either pinned to a material defect or jammed in place and unable to overcome the activation energy required to travel to the material surface or reach an oppositely charged monopole to annihilate5,27,28.

Examinations of skyrmions using 2D imaging have revealed a myriad of in-plane skyrmion string deformations including elongated structures13 and bent skyrmion strings that terminate on the surface29 or form at edges28, while three-dimensional (3D) imaging in confined systems has revealed axial modulation of skyrmion tubes30. In all of these instances, the physical systems being probed are subject to constricted geometries by virtue of the thin plate or needle-shaped samples required by the techniques. This introduces confinement effects, causing the system to exhibit drastically different energetics to those of bulk samples, favouring surface pinning31 and edge interactions28,32,33, which affect skyrmion formation, shape, interactions, dynamics and stabilization. Recent advances in the imaging and reconstruction of 3D magnetization vector fields have been made using X-ray magnetic microscopy34,35 and nanotomography36,37 techniques, exploiting dichroism in magnetic scattering and element-specific absorption contrast. However, such methods maintain low penetration depths, bounded material thicknesses and compositions and require vacuum-compatible setups that typically limit angular sampling37. A recent study38 confirmed the presence of interrupted and merging-type skyrmion strings in a sparsely populated micrometre-sized thin plate needle-shaped sample of Mn1.4Pt0.9Pd0.4Sn using scalar magnetic X-ray tomography. The observations, however, are limited to individual skyrmion strings in a sample thickness only a few times the skyrmion tube diameters, whose confined geometry and thickness gradient fundamentally alters the skyrmions shape and behaviour; bulk lattice skyrmion behaviour has yet to be experimentally observed.

Small-angle neutron scattering (SANS) is sensitive to bulk scattering features, enabling studies of truly bulk skyrmionic systems along different paths through the sample’s phase diagram. In particular, spatially averaged depth information may be obtained by forming rocking curves, integrating over one or more diffraction peaks as a function of sample angle, to extract the longitudinal correlation length of a skyrmion lattice4. The effects of recent external magnetic field and temperature history on jammed states has been shown using SANS, where skyrmion lattice defect densities were reduced using an ordering sequence where the external magnetic field is rocked relative to the sample39. Rocking the sample in the field39,40 produces skyrmion lattice reorientations that disentangle the jammed state, promoting additional skyrmion formation39,40, unless the segmentation defect creation barrier is too high or the pinning potential is too strong to liberate existing defects14,40,41.

Whereas previous studies are confined to surface-level techniques, confined systems or integral far-field SANS measurements that cannot produce real-space representations of the sample, here we perform tomography of the thermal equilibrium triangular skyrmion lattice phase to generate a 3D mean scattering feature reconstruction (MSFR) of a bulk skyrmion lattice. While existing neutron scattering tomography techniques have applied phase-retrieval algorithms to multiprojection ultra-SANS datasets to reconstruct 2D scattering length density maps of periodic structures42, the technique presented here takes somewhat of a divergent approach, coupling the angular projection datasets with a free-energy regularization. In particular, our reconstruction algorithm consists of first forming an estimator of a multiprojection set of SANS measurements by operating on the incoming neutron state with a forward operator, which takes the MSFR volume as its main input. Next, the sum of weighted residuals between the estimated and measured SANS patterns for multiple projections is minimized with respect to the MSFR. Without integrating over peak areas before forming the rocking curve, the weighted sum of measurement residuals χ2 is sensitive to shifting peak locations, shapes, intensities and correlations. The number of free parameters depends on the chosen MSFR volume but will usually be larger than the number of data points in the set of SANS images. The large degeneracy of possible solutions and the danger of overfitting the data is overcome by adding a free-energy regularizing functional to the objective function \(f={\chi }^{2}+\beta {{{\mathcal{F}}}}\), where \({{{\mathcal{F}}}}\) is the free energy of the MSFR and β is a Lagrange multiplier reminiscent of a Boltzmann factor. The free energy includes the Heisenberg exchange, Dzyaloshinskii–Moriya exchange and external field Zeeman terms and could be made to include additional interactions that are functionals of the spin density, although such terms are beyond the scope of the present work. Knowledge of the functional form of the free energy and average magnetization of the phase is therefore required for the application of this technique to a magnetic system. Similar techniques are often used in traditional computed tomography algorithms, in which case the total variation can be used as a regularizing functional43. The resulting MSFRs may be interpreted as containing the types of structures, and their densities, which are common within the sample. However, there is no portion of the sample that looks exactly like a MSFR, and there are a large number of possible MSFRs that would converge on a minimum of the objective function. Moreover, since the reconstructed magnetization volumes are representative of the bulk, they will not produce surface specific features (such as surface twisting24,44) unless it is frequent throughout the sample, comprising a sizable volume fraction of the magnetic features. One can also view minimizing the objective function as performing a micromagnetic simulation with the χ2 providing the local interaction and pinning potential terms in the free energy that cause lattice defects, thereby enforcing the lattice correlation lengths and structure encoded in the SANS patterns.

Multiprojection SANS measurements were made (Fig. 1) using an above room temperature skyrmion host Co8Zn8Mn4 previously characterized in ref. 40. This class of materials has been studied across a variety of techniques including SANS3,40,41,45,46,47, magnetic susceptibility3,40,45,46 and Lorentz transmission electron microscopy3,45. Given the disordered nature of the thermal equilibrium triangular lattice skyrmion phase as explored in refs. 40,41, a skyrmion ordering sequence in which the sample was rotated in the magnetic field was first applied to precipitate ordered and oriented skyrmion lattices (Methods). Collection of the tomographic angular projections was then achieved by rotating the sample and magnetic field coincident about the vertical axis specified in Fig. 1. The determination of the range and number of projections that should be collected for a given magnetization volume, or sample, is set by the correlation lengths of the phase. In the ideal case, the angular range should be sufficiently large so as to capture the entire decay of the rocking curve. Rotations past this point do not enhance the data unless there exist large off-axis domain structures or spin geometries in which the propagation wavevector lies parallel to the incident neutron direction. For example, conical domains may be probed by collecting angular projections about the perpendicular sample-field geometry; however, these are left as the subject of our future experimental explorations. For the case of highly disordered states characterized by long rocking curves with persistent non-zero angular intensities, this can lead to large angular ranges for the tomographic datasets, and can also present challenges for the rotation of cumbersome cryomagnets in the limited sample space available on typical SANS beamlines. In the experiment presented here, the ordered skyrmion sample and magnetic field were rotated through 30 projections over a 5.8° angular range, limited by the geometry of the guide field coils (Fig. 2a). Application of the ordering sequence seeks to both enhance the longitudinal and transverse correlation lengths of the phase while precipitating an oriented monodomain skyrmion lattice. Therefore, the ordering sequence should refocus intensity to the central peak of the rocking curve, while minimizing its full width at half-maximum and any potential persistent non-zero angular intensity offsets arising from misoriented chiral domains. As a result, the initial application of the ordering sequence enforces our measured angular range, while some of the rocking curve extrapolation performed at lower wave vectors, Q, gets filled in by the average magnetization parameter, 〈mz〉, and free-energy term. To determine the implications for how a limited-projection dataset would affect our results, we reconstructed digital phantoms under both ideal conditions, where SANS data were simulated for 31 projections between ±15° about both axes (no compound rotations) and the actual experimental projections (see Methods and the ideal versus limited-projection datasets in the Supplementary Information).

Fig. 1: Reconstruction technique applied to a bulk skyrmion material.
figure 1

Schematic outlining the reconstruction technique, beginning with the collection of experimental SANS angular projections (top) and ending with the output MSFR (bottom). The experimental sample-field geometry used during tomographic rotations is shown, where the sample and magnetic field were rotated coincident about an angle θ ensuring a fixed magnetic orientation parallel to the surface normal of the sample. The rocking curve is plotted for one of the horizontal peaks, showing the measured and extrapolated regions of the Lorentzian curve. Estimated and measured SANS patterns are shown for the initial and final MSFR, including their weighted residuals and minimization of the cost function. The reconstruction is shown for the entire 12.3 μm3 (1.8 μm × 1.8 μm × 3.7 μm) MSFR volume, with blue contours outlining the skyrmion tubes for an out-of-plane magnetization, mz = − 0.5. Enlarged 2D skyrmion spin textures are shown for three depths along the reconstruction with highlighted emergent magnetic charges. a.u., arbitrary units.

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Fig. 2: Magnetization slices of reconstructed experimental and phantom datasets.
figure 2

a–f, Experimental setup (a), measured SANS image (b) reproduced from ref. 40, minimized weighted residuals (c), reconstructed SANS image (d), and xy (e) and xz (f) slices of the MSFR. g–l, The corresponding plots for the phantom B limited-projection dataset (see highlighted row in Table 1) where g is the simulated input magnetization, h is the simulation SANS pattern and i–l are the same as in c–f, respectively. All SANS images are shown for the zero projection, with the guide field and sample aligned parallel to the neutron beam. The xy and xz magnetization slices are shown for a 1.8 μm × 1.8 μm area of the MSFR, showing the full in-plane magnetization slice and half of the longitudinal magnetization slice.

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2D slices of the measured and reconstructed SANS images, weighted residuals and MSFRs are shown for the bulk sample and phantom set B with average magnetizations of 0.42 and 0.41 (Fig. 2), respectively. Videos showing magnetization and topological defects as a function of depth for the sample MSFR, phantom B and its corresponding limited-projection and ideal MSFRs are available as Supplementary Videos 1–4. The full set of videos for every dataset listed in Table 1 is available at the online repository found in ref. 48. Since the reconstruction technique does not provide a one-to-one spatial mapping to any volume within the sample, deviations between Fig. 2g,k are expected. However, comparison between the two in-plane slices shows the presence of similar topological structures, transitions and skyrmion packing densities. A more accurate and comprehensive comparison entails looking at the entire input and MSFR volumes for average features and their densities. Average magnetization, topological saturation and defect densities are compared in Table 1. Phantoms were generated with the external field near the helical–skyrmion boundary (phantom A), the ideal value (phantom B) and the ferromagnetic–skyrmion boundary (phantom C). The choice of Zeeman term h was set such that the average magnetization of the reconstructions matched those estimated through magnetic susceptibility measurements. The required weighting of the Zeeman term h to achieve the average magnetization that was estimated from d.c. susceptibility was affected by the value of χ2. The favoured MSFRs are thus those with values of h that produce the estimated average magnetization and they are highlighted in Table 1. The agreement between phantom and MSFR is reasonable for phantoms B and C, with deviations between ideal and limited-projection datasets probably arising owing to the truncated rocking curve of the limited datasets. However, the MSFR for phantom A overestimates the branching defect density. This is probably attributable to the seeding, where phantom A transitioned from an average magnetization of 〈mz〉 = 0.4 to 〈mz〉 = 0.33 during the free-energy relaxation (Methods), while the MSFR had a net magnetization change of 〈mz〉 = 0.35 to 〈mz〉 = 0.32, suggesting hysteresis-like effects can impact the fidelity of the reconstructions. Such effects are reminiscent of real samples, wherein sample histories and trajectories through phase space may alter the stability, chiral and topological volume fractions and defect densities of the phase49,50,51. Future studies may incorporate variations of the estimated 〈mz〉 parameter to simulate and incorporate these hysteresis effects.

Table 1 Zeeman term weight in reduced field units h, seeded average magnetization 〈mz〉seed, final average magnetization 〈mz〉, topological saturation \({N}_{{{{\rm{sk}}}}}/{N}_{\max }\) and defect densities ρ of the phantoms and associated MSFRs and sample MSFR
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The sample MSFR topological saturation \({N}_{{{{\rm{sk}}}}}/{N}_{\max }\) is found to be 63% of that of a perfect skyrmion triangular lattice with the same reciprocal space scattering location, Q0. This is reflected by the presence of transverse and longitudinal lattice distortions. In particular, skyrmion voids, bimerons and elongated spin structures present in the 2D MSFR magnetization slices reduce the number of skyrmions from that of an ideal hexagonally packed lattice. Similarly, interruption of the skyrmion strings along their length may be visualized by taking an x–z slice of the MSFR as shown in Fig. 2f.

3D visualizations of portions of the sample MSFR show skyrmion nucleation and annihilation along their depth, where segmentation S± and branching B± (anti)monopoles mediate the change in topology at skyrmion transition points (Fig. 3e). These features are similar to those observed for simulations of skyrmion annihilation in three dimensions through helical and conical pathways13,26,27. Notably, some branching events are observed to occur along wave vectors Q that are offset by 60° from the horizontal nearest-neighbouring skyrmion, producing a skyrmion twisting effect (Fig. 3a). Conversely, some instances of segmentation events exhibit pairs of S± (anti)monopoles that cup skyrmions, producing spatially localized skyrmion filaments that extend longitudinally over a few lattice periods (Fig. 3d) and are reminiscent of magnetic torons52. Given the field history of the sample, wherein saturation to the ferromagnetic phase was performed before collection of the tomography data, these structures may represent a skyrmion survival mechanism in the field-polarized state via magnetic torons on defect pinning sites41. Future studies will be performed to examine the prevalence of these structures as a function of field history and average magnetization, providing insight into skyrmion elongation and stabilization mechanisms.

Fig. 3: 3D Skyrmion tubes and their topological transition defect pathways.
figure 3

Characteristic topological transitions present in the MSFR showing branching (B) and segmentation (S) (anti)monopole pathways. Skyrmion topological transitions accompanied by a branching monopole-antimonopole pair (a), segmentation monopole-antimonopole pair (b), combination of segmentation monopole and branching antimonopole (c), and a localized skyrmion filament structure cupped by a segmentation monopole-antimonopole pair (d). The blue contours outline the skyrmion tubes where the out-of-plane magnetization mz = −0.5. Some of the skyrmion tubes are masked to highlight the regions undergoing changes in topology. Emergent magnetic charge density contours are shown for the two types of skyrmion topological transition processes with S+, S−, B+ and B− corresponding to purple, orange, red and green, respectively. Branching and segmentation emergent (anti)monopoles are observed to occur at transition points along the tubes, displaying distinct spin textures determined by the sign of ∂zmz. e, Enlarged spin textures are shown for the labelled branching and segmentation (anti)monopoles with each spin corresponding to one voxel. All skyrmion features are shown for a depth of 37 voxels, equating to 531 nm (Methods).

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The energy source required for emergent (anti)monopole creation may derive from internal chemical disorder present in the material (such as in material imperfections or site disorder9,14,26), thermal activation53,54,55 and the external magnetic field setting relative to the helical and ferromagnetic phase boundaries13,29,54. For the case of the magnetic field setting, segmentation and branching (anti)monopoles are thought to control skyrmion annihilation upon increasing and decreasing fields, respectively13,27,56. This is reinforced by reconstructions performed on equivalent datasets under different average magnetization and reduced field h conditions; a shifting prevalence from branching defects in the low-field case to segmenting defects in the high-field case is observed.

The preferred sample MSFR shows segmentation and branching (anti)monopoles seemingly jammed in place and unable to travel to the sample surfaces or annihilate. The observed defect densities could be due to pinning potentials in the material that would trap the (anti)monopoles and inhibit their motion14. In this case, the prevalence of these structures may indicate the degree of internal chemical disorder, providing snapshots of magnetic defect pinning centres. Alternatively, these features may be evidence of an incomplete ordering sequence, wherein the limited magnetic field directions during rotation did not allow the monopoles to propagate along enough angular paths in the sample to completely break up the jammed labyrinth domains39. Future SANS tomography measurements taken as a function of skyrmion ordering could decouple these two possibilities. Implementation with structured neutron beams, which possess an analogous non-trivial winding character in spin57 or phase58,59, may provide estimates of the magnetic defect densities based on the vertical widening of the skyrmion peaks in the transverse geometry. Alternatively, the defects themselves may be viewed as the magnetic equivalent of phase singularities, capable of generating topological neutron states for probing nanometric sample topologies.

We have shown experimental visualizations of the 3D character of a bulk skyrmion lattice through MSFR, generated via a SANS tomography technique. The present results uncover the stabilization and evolution of a triangular skyrmion lattice in the bulk through 3D topological transitions that exhibit a mixture of distinctive segmenting and branching (anti)monopole defects. Exotic features for these two event types are observed in the MSFR, characterized by skyrmion twisting during branching events and localized skyrmion filament structures cupped by S± (anti)monopoles. Our SANS tomography techniques provide unique insights into skyrmion formation, annihilation and transition processes through (anti)monopole defects, opening the door to future studies of bulk micromagnetic materials on unprecedented length scales, including skyrmion and emergent (anti)monopole structure, behaviour and dynamic processes.

Methods

The Co8Zn8Mn4 material was grown via a modified Bridgman technique in a rapid high-temperature furnace at McMaster University, before being cut into a cube of dimensions 3.4 mm × 3.3 mm × 3.0 mm and oriented such that the (100) direction was coming out of the major face. Details of the sample, as well as single-projection SANS and susceptibility measurements, are available elsewhere40.

Multiprojection, unpolarized SANS was performed at the NG7-30m beamline at the National Institute for Standards and Technology (NIST) for a 15 m beam configuration at a wavelength of 6 Å. The sample was field cooled from 420 K in a field of 250 G to 310 K. Upon entering the thermal equilibrium triangular lattice skyrmion phase a uniform ring manifested in SANS, indicative of a jammed chiral state composed of misoriented skyrmion, labyrinth and helical domains41. The skyrmion sample was rotated symmetrically in the magnetic field 10 times over a ± 60° angular range to achieve the ordered six-fold pattern shown in Fig. 1. From here, multiprojection SANS data were collected for a static skyrmion configuration, rotating the sample and field together on the same rotation stage about the vertical axis specified in Fig. 1 to ensure the magnetic field orientation remained fixed—with the field direction parallel to the surface normal of the sample. These tomographic rotations were performed for 30 angles from −2.9° to 2.9°. Incorporating additional tomography axes may enhance the data, particularly in the case of sample tilting for skyrmion states with very short longitudinal correlation lengths; however, this is left for our future studies.

Digital phantoms were made by seeding a log-likelihood gain relaxation using Ubermag60 with a high-energy lattice made to have a fixed correlation volume. The log-likelihood gain relaxation was stopped after ten iterations to retain a reasonable density of defects. The system used the free-energy functional

$$E[{{{\mathbf{m}}}}]=-{A}{{{\mathbf {m}}}}\cdot {\nabla }^{2}{{{\mathbf{m}}}}+D{{{\mathbf{m}}}}\cdot (\nabla \times {{{\mathbf{m}}}})-{\mu }_{0}{M}_{\mathrm{s}}{{{\mathbf{H}}}}\cdot {{{\mathbf{m}}}}$$
(1)

with exchange stiffness A = 10 pJ m−1, Dzyaloshinskii–Moriya constant D = 3.93 × 10−3 J m−1, saturation magnetization Ms = 1 × 106 A m−1 and external field values of H = 0.2080 T, 0.2780 T and 0.3475 T for phantoms A, B and C, respectively.

The seed was generated by alternating between Fourier-space and real-space constraints. The magnitude of the sample magnetization in Fourier space is set by desired correlation volume, the real-space magnitude of the sample magnetization was constrained to unity everywhere and the transverse magnetization was set to be in the direction of the curl of the longitudinal magnetization.

SANS patterns were simulated by applying a forward operator to a randomly selected and randomly translated phantom of 20 phantoms made with differing random initialization. The resulting SANS images produced by repeating this process at least 100 times were averaged to simulate an incoherent neutron source and create the resulting simulated multiprojection SANS data.

The voxel size of the MSFR is determined by the Fourier-space span of the SANS images dx = 2π/Qtot = 2π/(dQ N). In this experiment dx = 14.3 nm, as determined by the resolution dQ = 3.4 μm−1 and size N × N = 128 × 128 of the SANS images. The height of the MSFR was set to 256 voxels, for a total volume of 1.8 μm × 1.8 μm × 3.7 μm = 12.3 μm3.

To compute the reconstructions, the cost function was minimized using a conjugate gradient method. The m2 = 1 constraint was enforced by defining search directions in terms of the angular fields \(\varTheta =\arcsin ({m}_{z})\) and \(\varPhi =\arctan ({m}_{y},{m}_{x})\). Ten iterations were run with equal weights for all residuals, which can aid in convergence when measurement uncertainties are dominated by counting statistics and there are regions of low count rates. Following the 10 iterations, 100 iterations were run with the weights given the measurement uncertainties provided by the SANS reduction software61. A maximum weight was introduced for the low count rate regions to aid in convergence. The resulting average magnetization was found to depend on the weight given to the Zeeman term h, so reconstructions over a range of h and starting average magnetization 〈mz〉 were performed (Table 1), with the preferred MSFR being the one that most closely matched the phantom or experimental average magnetization.

Details on the forward propagator, gradient and defect density calculations are provided in the Supplementary Information.