Micromagnetic simulationsThe micromagnetic way used to be adopted on this paintings. The full power of the gadget comprises the alternate power, the DMI power, the Zeeman power and the power of the demagnetizing fields45:$$start{array}{l}{mathcal{E}},=,{int }_{{V}_{{rm{m}}}}{rm{d}}{bf{r}}{mathcal{A}}sum _{i=x,y,z}| nabla {m}_{i} ^{2}+{mathcal{D}},{bf{m}},cdot (nabla ,occasions ,{bf{m}})-{M}_{{rm{s}}},{bf{m}},cdot ,{bf{B}}+ ,,,,+frac{1}{2{mu }_{0}}{int }_{{{mathbb{R}}}^{3}}{rm{d}}{bf{r}}sum _{i=x,y,z}| nabla {A}_{{rm{d}},i} ^{2},finish{array}$$
(1)
the place m(r) = M(r)/Ms is a unit vector area that defines the route of the magnetization, Ms = ∣M(r)∣ is the saturation magnetization and μ0 is the vacuum permeability (μ0 ≈ 1.257 μN A−2). The constants ({mathcal{A}}) and ({mathcal{D}}) are the alternate stiffness and the isotropic bulk DMI, respectively. The ratio between ({mathcal{A}}) and ({mathcal{D}}) defines the equilibrium length of the conical part, ({L}_{{rm{D}}}=4pi {mathcal{A}}/D). The magnetic area in equation (1), B = Bext + ∇ × Advert, is the sum of the exterior magnetic area and the demagnetizing area, the place Advert(r) is the part of the magnetic vector doable precipitated via the magnetization. For the calculations within the bulk gadget, we set the exterior magnetic area ({{bf{B}}}_{{rm{ext}}}=0.5,{B}_{{rm{D}}}{widehat{{bf{e}}}}_{z}), the place ({B}_{{rm{D}}}={{mathcal{D}}}^{2}/(2{M}_{{rm{s}}}{mathcal{A}})) is the conical part saturation area within the absence of demagnetizing fields20. We used the next subject matter parameters for FeGe20,33: ({mathcal{A}}) = 4.75 pJ m−1, ({mathcal{D}}) = 0.853 mJ m−2 and Ms = 384 kA m−1. For the 0.5-μm-diameter and 180-nm-thick disk pattern depicted in Fig. 1a–e, the calculations had been carried out on a mesh with 256 × 256 × 64 cuboids. Calculations for the 1 μm × 1 μm × 180 nm pattern had been carried out on a mesh with 400 × 400 × 72 cuboids. For the majority magnet, we exclude dipole–dipole interactions and imagine a site of measurement 5LD × 5LD × 10LD underneath periodic boundary prerequisites on a mesh with 256 × 256 × 512 cuboids.Following the arguments introduced in a prior study35, a skinny floor layer of the isotropic chiral magnet crystal is broken throughout FIB milling and will also be successfully approximated via subject matter parameters which might be just like the ones of the majority crystal, however with the DMI coupling consistent set to 0. Within the earlier report35, the thickness of the FIB-damaged layer of an FeGe nanocylinder used to be estimated to be 6 ± 1 nm. In keeping with every other report21, the thickness of the broken layer of an FeGe needle-like pattern is round 10 nm. Right here, we think an intermediate thickness for the broken layer of seven.5 nm (corresponding to a few floor cuboids).It will have to be famous that the presence or absence of a broken layer in our simulations has nearly no impact at the steadiness of the answers proven in Fig. 4. The distinction in theoretical Lorentz TEM pictures in Fig. 3 additionally does now not exchange considerably when the presence of a broken layer is omitted. On the other hand, the presence of a broken floor layer has a an important position in hopfion-ring nucleation. Within the simulations, throughout the utility of a magnetic area within the adverse and certain instructions with recognize to the z axis, we simplest succeeded in staring at hopfion-ring nucleation, as proven in Fig. 1a–d, within the presence of a broken floor layer.Statically solid answers of the Hamiltonian (equation (1)) had been discovered via the usage of the numerical power minimization approach described previously20 the usage of the Excalibur code46. The answers had been double-checked the usage of the publicly to be had tool Mumax47. Within the Supplementary data, we additionally supply 3 Mumax scripts, which can be utilized to breed the result of our micromagnetic simulations. Script I permits the hopfion-ring nucleation depicted in Fig. 1a–d to be reproduced. Since the states depicted in Fig. 1a,d are two states with other energies which might be stabilized in similar prerequisites, the transition between them calls for further power pumping. The power stability between those states relies on the carried out area. Within the experimental set-up, this in-field transition is enhanced via thermal fluctuation and, consequently, has a probabilistic personality. To make the nucleation of the hopfion ring deterministic (reproducible), within the micromagnetic simulations we use an abrupt transfer of the magnetic area (with a step of round 100 mT) to conquer the barrier between the metastable states. Script II, with minor adjustments of the preliminary states mentioned within the subsequent segment, can be utilized to breed the states proven in Fig. 4. For an outline of Script III, see the next segment.Preliminary state for hopfion rings in micromagnetic simulationsOn the foundation of experimental observations and theoretical research, we spotted that the presence of the conical part round other localized states leads to an extra contribution to the electron optical part shift that adjustments across the perimeter of the pattern. To acquire the magnetic textures in nanoscale samples, we used preliminary configurations comparable to a superposition of cylindrical domain names, with their magnetization pointing up and down, embedded in a conical part and with an extra part modulation similar to a vortex within the xy aircraft of the shape$$Theta ={rm{acos}}left(frac{{B}_{{rm{ext}}}}{{B}_{{rm{D}}}+{mu }_{0}{M}_{{rm{s}}}}appropriate),,Phi ={rm{atan}}frac{y}{x}+frac{pi }{2}+kz,$$
(2)
the place ok = 2π/LD is the wave quantity. In every other study27, identical vortex-cone configurations had been mentioned within the context of screw dislocations in bulk chiral magnets. Right here, a magnetic configuration approximated via equation (2) seems owing to an interaction between short-range interactions (Heisenberg alternate and DMI) and a long-range demagnetizing area. This impact has prior to now been seen in samples of confined geometry26,32,33.Consultant examples of 2 preliminary states are illustrated in Prolonged Knowledge Fig. 9a. Solid magnetic states bought from those preliminary states after power minimization are proven in Prolonged Knowledge Fig. 9b,c. The state with a compact hopfion ring now not simplest has decrease power, but additionally supplies distinction in theoretical Lorentz TEM pictures that correctly fit experimental pictures (Prolonged Knowledge Fig. 9e,f). The effects proven in Prolonged Knowledge Fig. 9 will also be reproduced via the usage of Mumax Script II.For simulations of the majority, skyrmion strings with hopfion rings had been embedded into the uniform conical part:$$Theta ={rm{acos}}left(,{B}_{{rm{ext}}}/{B}_{{rm{D}}}appropriate),,Phi =kz.$$
(3)
For hopfion rings, we used the next toroidal ansatz:$$Theta =pi ,left(1-frac{eta }{{R}_{1}}appropriate),,,0le eta le {R}_{1},$$
(4)
$$Phi ={rm{atan}},left(frac{y}{x}appropriate)-{rm{atan}},left(frac{z}{{R}_{2}-rho }appropriate)-frac{pi }{2},$$
(5)
the place R1 and R2 are the minor and main radii of a torus, respectively, (rho =sqrt{{x}^{2}+{y}^{2}}) and (eta =sqrt{{({R}_{2}-rho )}^{2}+{z}^{2}}). We additionally refer the reader to the Mumax Script III for preliminary state implementation. By means of default, Script III can reproduce a fancy configuration within the bulk gadget, as proven in Prolonged Knowledge Fig. 8h. With minor adjustments, it will also be used to copy all different states.Simulations of electron optical phase-shift and Lorentz TEM imagesBy the usage of the part object approximation and assuming that the electron beam is antiparallel to the z axis, the wave serve as of an electron beam will also be written as follows48:$${Psi }_{0}(x,y)propto exp left(ivarphi (x,y)appropriate),$$
(6)
the place φ(x, y) is the magnetic contribution to the part shift49$$varphi (x,y)=frac{2pi e}{h}underset{-infty }{overset{+infty }{int }},{rm{d}}z,{{bf{A}}}_{{rm{d}}}cdot {widehat{{bf{e}}}}_{{rm{z}}},$$
(7)
e is an basic (certain) fee (round 1.6 × 10−19 C) and h is Planck’s consistent (roughly 6.63 × 10−34 m2 kg s−1). As a result of our way for the answer of the micromagnetic drawback recovers the magnetic vector doable Advert, simulation of the electron optical part shift is easy.Within the Fresnel mode of Lorentz TEM, neglecting aberrations as opposed to defocus, aperture purposes and resources of incoherence and blurring, the wave serve as on the detector aircraft will also be written within the shape$${Psi }_{Delta z}(x,y)propto int ,int ,{rm{d}}{x}^{{high} }{rm{d}}{y}^{{high} },{Psi }_{0}({x}^{{high} },{y}^{{high} })Ok(x-{x}^{{high} },y-{y}^{{high} }),$$
(8)
the place the kernel is given via the expression$$Ok(xi ,eta )=exp left(frac{ipi }{lambda Delta z}({xi }^{2}+{eta }^{2})appropriate),$$
(9)
the relativistic electron wavelength is$$lambda =frac{hc}{sqrt{{(eU)}^{2}+2eU{m}_{e}{c}^{2}}},$$
(10)
Δz is the defocus of the imaging lens, c is the velocity of sunshine (roughly 2.99 × 108 m s−1), U is the microscope accelerating voltage and me is the electron leisure mass (round 9.11 × 10−31 kg). The picture depth is then calculated the usage of the expression$$I(x,y)propto | {Psi }_{Delta z}(x,y) ^{2}.$$
(11)
For extra information about the calculation of Lorentz TEM pictures, see ref. 20.Homotopy-group analysisSkyrmion topological chargeFor magnetic textures localized within the aircraft space (Omega subseteq {{mathbb{R}}}^{2}), such that on the boundary of this space ∂Ω the magnetization area m(∂Ω) = m0, the classifying organization is the second one homotopy organization of the gap ({{mathbb{S}}}^{2}) on the base level m0, and there’s an isomorphism to the crowd of integers (Abelian organization with recognize to addition):$${pi }_{2}({{mathbb{S}}}^{2},{{bf{m}}}_{0})={mathbb{Z}}.$$
(12)
This signifies that any steady magnetic texture enjoyable the above standards of localization will also be attributed to an integer quantity, which is usually known as the skyrmion topological fee (or skyrmion topological index), and will also be calculated as follows:$$left{,start{array}{ll} & Q=frac{1}{4pi }{int }_{Omega }{rm{d}}{r}_{1}{rm{d}}{r}_{2},{bf{F}}cdot {widehat{{bf{e}}}}_{{r}_{3}}, & {{bf{m}}}_{0}cdot {widehat{{bf{e}}}}_{{r}_{3}} > 0,finish{array}appropriate.$$
(13)
the place$${bf{F}}=left(start{array}{l}{bf{m}}cdot [{partial }_{{r}_{2}}{bf{m}}times {partial }_{{r}_{3}}{bf{m}}] {bf{m}}cdot [{partial }_{{r}_{3}}{bf{m}}times {partial }_{{r}_{1}}{bf{m}}] {bf{m}}cdot [{partial }_{{r}_{1}}{bf{m}}times {partial }_{{r}_{2}}{bf{m}}]finish{array}appropriate)$$
(14)
is the vector of curvature50,51, which could also be identified (as much as a prefactor) because the gyro-vector or vorticity10,52,53,54, and r1, r2 and r3 are native right-handed Cartesian coordinates.The unit area m will also be parameterized at the ({{mathbb{S}}}^{2}) sphere the usage of polar and azimuthal angles Θ and Φ, respectively, within the shape ({bf{m}}=(cos Phi sin Theta ,sin Phi sin Theta ,cos Theta )). The corresponding topological invariant, as much as the signal, is the stage of mapping of the skyrmion localization space onto the sphere55, which will also be calculated the usage of the highest a part of equation (13), assuming that r1 and r2 lie within the skyrmion aircraft. It will have to be famous that the signal of the integral within the best a part of equation (13) relies on the number of the orientation of the coordinate gadget. As an example, in Fig. 4a the signal of Q relies on whether or not the r3 axis is parallel or antiparallel to the z axis and equals − 11 or 11, respectively. The situation within the backside a part of equation (13) gets rid of this ambiguity. A justification for this observation, in line with the idea of elementary invariants, will also be present in a prior study56. The native coordinate gadget (r1, r2, r3) for calculating the topological fee Q of a specific skyrmion is selected in line with the situation within the backside a part of equation (13).For skyrmions that experience other m0 within the world coordinate gadget, equation (12) isn’t globally appropriate since the base points57 are other. On the other hand, isomorphisms to the crowd of integers can at all times be finished thru steady particular person transformations of vector fields to compare the vectors m0 to 1 base level.Right here, we use the similar conference for the signal of the topological fee as earlier reports20,28,29,58, such that an basic Bloch-type or Neel-type skyrmion has Q = −1.Hopfion topological chargeFor a magnetic texture localized throughout the 3-D area (Omega subseteq {{mathbb{R}}}^{3}), with a hard and fast magnetization m(∂Ω) = m0 on the boundary ∂Ω of the area, the classifying organization corresponds to the 3rd homotopy organization of the gap ({{mathbb{S}}}^{2}) on the base level m0:$${pi }_{3}({{mathbb{S}}}^{2},{{bf{m}}}_{0})={mathbb{Z}}.$$
(15)
The corresponding topological fee, which is referred to as the Hopf invariant, will also be calculated the usage of Whitehead’s formula51,59:$$H=-frac{1}{16{pi }^{2}}{int }_{Omega }{rm{d}}{r}_{1}{rm{d}}{r}_{2}{rm{d}}{r}_{3},{bf{F}}cdot [{(nabla times )}^{-1}{bf{F}}].$$
(16)
Skyrmion–hopfion topological chargeTo analyse the continual texture localized on a section of a skyrmion string, we use the compactification way and different strategies of algebraic topology60. First, we notice that, owing to the invariance of Q alongside a skyrmion string, the decrease and higher cross-sections bounding the skyrmion string section are similar via a trivial transformation. This signifies that the measurement alongside the skyrmion string will also be compactified to a circle ({{mathbb{S}}}^{1}). 2d, we notice that the conical part and the part with uniform magnetization m0 are identical to one another as much as a trivial transformation. By means of exploiting this statement, one can compactify the remainder two dimensions. The magnetic texture localized on a section of a skyrmion string will also be handled as whether it is confined inside of a forged torus (Omega ={{mathbb{D}}}^{2}occasions {{mathbb{S}}}^{1}). The noncollinearities of m are then localized within Ω, whilst in all places on its floor (partial Omega ={{mathbb{S}}}^{1}occasions {{mathbb{S}}}^{1}) the magnetization m(∂Ω) = m0 is fastened.Thereby, the homotopy classification arises from a continual map from a one-point compactified forged torus to the spin house:$${{mathbb{D}}}^{2},occasions ,{{mathbb{S}}}^{1}/{{mathbb{S}}}^{1},occasions ,{{mathbb{S}}}^{1}to {{mathbb{S}}}^{2}.$$
(17)
By means of the usage of the homeomorphism of the quotient areas ({{mathbb{D}}}^{2},occasions ,{{mathbb{S}}}^{1}/{{mathbb{S}}}^{1},occasions ,{{mathbb{S}}}^{1}) and ({{mathbb{S}}}^{3}/{{mathbb{S}}}^{1}), in addition to the homotopy equivalence between ({{mathbb{S}}}^{3}/{{mathbb{S}}}^{1}) and ({{mathbb{S}}}^{2}wedge {{mathbb{S}}}^{3}), we discover a homotopy identical map$${{mathbb{S}}}^{2}wedge {{mathbb{S}}}^{3}to {{mathbb{S}}}^{2}.$$
(18)
Taking the bottom level, m0, as some degree commonplace to the wedge sum, we in an instant to find the homotopy organization$$G={pi }_{2}({{mathbb{S}}}^{2},{{bf{m}}}_{0})occasions {pi }_{3}({{mathbb{S}}}^{2},{{bf{m}}}_{0})={mathbb{Z}}occasions {mathbb{Z}},$$
(19)
the place π2 and π3 correspond to equations (12) and (15), respectively, and the parts of the topological fee are matter to equations (13) and (16), respectively. The topological index for the textures depicted in Fig. 4 and Prolonged Knowledge Fig. 8 then represents the ordered pair of integers (Q, H).Calculation of topological chargesTo compactify the textures bought in micromagnetic simulations, we used the nested field way. This system comes to solving and striking the field containing the studied texture on the centre of a rather better computational field. The computational field has periodic boundary prerequisites alongside the z route, and the remainder barriers are fastened ({{bf{m}}}_{0}={widehat{{bf{e}}}}_{z}). To verify continuity of the vector area m within the transition areas between the nested containers, we minimized the Dirichlet power, ∫dr∣∇ m∣2. Subsequent, to calculate F and the topological fee Q, we used a prior to now proposed lattice approach61. The vector doable of the divergence-free area F used to be bought via comparing the integral:$${(nabla occasions )}^{-1}{bf{F}}=int {rm{d}}x,{bf{F}}occasions {widehat{{bf{e}}}}_{x}.$$
(20)
The Hopf index H used to be then made up our minds via numerically integrating equation (16).For extra verification, we additionally computed the index H via calculating the linking quantity for curves in actual house that corresponded to 2 other issues at the spin sphere62.Derivation of 0 modeThe 0 mode is bought via analysing the symmetries of the Hamiltonian introduced within the supplementary subject matter of a prior report32. With out dipole–dipole interactions, the power density of the majority gadget in equation (1) is invariant underneath the next transformations from ({{bf{m}}}^{{high} }({{bf{r}}}^{{high} })) to m(r) and vice versa:$${bf{m}}({bf{r}})=left(start{array}{lll}cos (ok{z}_{0}) & -sin (ok{z}_{0}) & 0 sin (ok{z}_{0}) & cos (ok{z}_{0}) & 0 0 & 0 & 1end{array}appropriate)cdot {{bf{m}}}^{{high} }({{bf{r}}}^{{high} }),$$
(21)
the place$${{bf{r}}}^{{high} }=left(start{array}{rcl}cos (ok{z}_{0}) & sin (ok{z}_{0}) & 0 -sin (ok{z}_{0}) & cos (ok{z}_{0}) & 0 0 & 0 & 1end{array}appropriate)cdot {bf{r}}-{z}_{0}{widehat{{bf{e}}}}_{z},$$
(22)
and z0 is an arbitrary parameter, which, in essentially the most normal case, defines the screw-like movement of a whole magnetic texture in regards to the z axis with pitch 2π/ok = LD. Of notice, there are no less than two circumstances wherein the transformation (equations (21) and (22)) does now not impact the magnetic texture, which means that ({{bf{m}}}^{{high} }({{bf{r}}}^{{high} })={bf{m}}({bf{r}})) holds for any price of z0. The primary case is reasonably trivial and corresponds to the conical part with the wave vector aligned parallel to the z axis (see equation (3)). The second one case is especially intriguing, and comes to the skyrmion string within the conical part, wherein the principle axis of the string aligns with the rotation axis32. Making use of the transformation in equations (21) and (22) to the skyrmion string with a hopfion ring describes the screw movement of the hopfion ring across the string representing a nil mode. The parameter z0, on this case, denotes the displacement of the hopfion ring alongside the string. Supplementary Movies 7 and eight supply a visualization of this kind of screw movement of 2 other hopfion rings depicted in Fig. 4e,f. Proof for such 0 mode in different 3-D solitons will also be present in earlier reports8,63.Specimen preparationFeGe TEM specimens had been ready from a unmarried crystal of B20-type FeGe the usage of a FIB workstation and a lift-out method20.Magnetic imaging within the transmission electron microscopeThe Fresnel defocus mode of Lorentz imaging and off-axis electron holography had been carried out in an FEI Titan 60-300 TEM operated at 300 kV. For each tactics, the microscope used to be operated in Lorentz mode with the pattern to start with in magnetic-field-free prerequisites. The traditional microscope goal lens used to be then used to use out-of-plane magnetic fields to the pattern of between −0.15 and + 1.5 T. A liquid-nitrogen-cooled specimen holder (Gatan type 636) used to be used to change the pattern temperature between 95 and 380 Ok. Fresnel defocus Lorentz pictures and off-axis electron holograms had been recorded the usage of a 4k × 4k Gatan K2 IS direct electron counting detector. Lorentz pictures had been recorded at a defocus distance of 400 μm, except another way specified. A couple of off-axis electron holograms, each and every with a 4 s publicity time, had been recorded to beef up the signal-to-noise ratio and analysed the usage of a normal speedy Fourier turn into set of rules in Holoworks tool (Gatan). The magnetic induction map proven in Fig. 1 used to be bought from the gradient of an experimental magnetic part symbol.Supplementary Movies 1–5 display in situ Lorentz TEM pictures captured at a defocus distance of roughly 400 μm and at a pattern temperature of 180 Ok. Every video starts with a number of cycles of area swapping, wherein the carried out magnetic area alternates between certain and adverse instructions perpendicular to the plate. The sector amplitude is restricted to 50 mT or much less. Because the magnetic area within the transmission electron microscope is supplied via the target lens, this alternating area results in a visual rotation of the picture at the display. Counter-clockwise rotation signifies an building up within the area in opposition to the viewer and vice versa. Those field-swapping cycles induce edge modulations that propagate in opposition to the centre of the pattern. The main goal of this cycle is to generate edge modulations that propagate out of the unfastened edges. After a couple of field-swapping cycles, closed loops close to the centre of the sq. pattern shape. To improve visibility, the playback velocity of the entire movies has been doubled. As soon as no less than one closed loop has been shaped clear of the pattern edges, the magnetic area is higher to roughly 150 mT, ensuing within the nucleation of a hopfion ring. Supplementary Video 1 illustrates the nucleation of a double hopfion ring.Supplementary Video 4 finds instabilities of the hopfion ring. After hopfion-ring nucleation, the magnetic area used to be to start with lowered under a threshold price of roughly 50 mT, inflicting the hopfion ring to lose its form and elongate over the pattern. Therefore, the sphere used to be higher once more, resulting in the reformation of a compact hopfion ring surrounding six skyrmion strings. In any case, the sphere used to be higher additional above 190 mT, resulting in the cave in of the hopfion ring. The concluding body of Supplementary Video 4 depicts a cluster of six skyrmions with out a hopfion ring.
