Size-Dependence and High Temperature Stability of Radial Vortex Magnetic Textures Imprinted by Superconductor Stray Fields

Swirling spin textures, including topologically nontrivial states, such as skyrmions, chiral domain walls, and magnetic vortices, have garnered significant attention within the scientific community due to their appeal from both fundamental and applied points of view. However, their creation, controlled manipulation, and stability are typically constrained to certain systems with specific crystallographic symmetries, bulk or interface interactions, and/or a precise stacking sequence of materials. Recently, a new approach has shown potential for the imprint of magnetic radial vortices in soft ferromagnetic compounds making use of the stray field of YBa2Cu3O7-δ superconducting microstructures in ferromagnet/superconductor (FM/SC) hybrids at temperatures below the superconducting transition temperature (TC). Here, we explore the lower size limit for the imprint of magnetic radial vortices in square and disc shaped structures as well as the persistence of these spin textures above TC, with magnetic domains retaining partial memory. Structures with circular geometry and with FM patterned to smaller radius than the superconductor island facilitate the imprinting of magnetic radial vortices and improve their stability above TC, in contrast to square structures where the presence of magnetic domains increases the dipolar energy. Micromagnetic modeling coupled with a SC field model reveals that the stabilization mechanism above TC is mediated by microstructural defects. Superconducting control of swirling spin textures, and their stabilization above the superconducting transition temperature by means of defect engineering holds promising prospects for shaping superconducting spintronics based on magnetic textures.


Introduction
The rise of spintronics has stimulated the interest in topologically non-trivial spin configurations, such as skyrmions [1][2][3] , merons [4][5][6] , and magnetic (radial) vortices [7][8][9] , promoting the search for new materials, methods, and/or configurations in which these structures can be created, stabilized, and controlled.There have been significant advances in our understanding of the physics governing the formation of these non-trivial magnetic domain configurations.However, a method for creating and stabilizing complex spin textures, and applicable to a large variety of compounds is yet missing.
Recently, a new approach based in the use of hybrid superconductor/ferromagnet (SC/FM) microstructures has demonstrated the possibility to generate and control swirling spin textures [10,11] .Superconductivity and ferromagnetism are two electronic ground states which despite their antagonistic character, may become synergistic in superconductor/ferromagnet hybrids.They yield exciting responses as, for example, the recently demonstrated long range supercurrent and Josephson effects driven by equal spin triplet superconducting correlations [11][12][13] which coexist peacefully with ferromagnetism.
A wide category of effects in SC/FM hybrids involve the influence of the ferromagnet on the superconducting ground state.This influence is mediated by the stray fields of ferromagnetic domains or magnetic chiral structures (like magnetic vortices, skyrmions or domain walls) on the dissipation properties and critical current characteristics of the superconductor [14][15][16][17][18][19][20] .For instance, the presence of a magnetic vortex in the ferromagnetic barrier of a Josephson junction can tailor the supercurrent pathways making it to behave as a SQUID [21] or 0-π SQUID [22] .
The opposite, that is the role played by superconducting stray fields in the magnetic ground state of ferromagnets, has comparatively received less attention.To this regard, Palau et al. [23] and Sander et al. [10] have shown the possibility of crafting swirling spin textures in ferromagnetic systems by making use of the magnetic stray fields generated by the trapped flux in structured type II superconductors.
Below the superconducting transition temperature, the application and removal of an out-of-plane magnetic field yields the generation of screening supercurrents due to the penetration, pinning and expulsion of magnetic flux quanta [24,25] .Within the mixed state, these supercurrents flow following the geometrical contour of the SC structure [26][27][28] , with a geometry and sense of rotation which depend on magnetic history [10,23] .Supercurrents present after removal of the external magnetic field, give rise to a stray magnetic field whose strength and direction varies locally [29] , see Supplementary Information Section 1.In ferromagnetic systems with perpendicular magnetic anisotropy (PMA), the out-of-plane component of the superconductor stray-field can be employed to imprint unusual magnetic textures [10] .The imprint is stabilized by the PMA so that it remains even when supercurrents have vanished for temperatures above TC.On the other hand, Palau et al. showed that in ferromagnetic layers with in-plane the in-plane components of the SC stray-field can be utilized to imprint magnetic domain distributions akin to radial vortices with a lateral size of 20 µm [23] .In these, the in-plane magnetization can point towards or away from the core along radial directions orthogonal to the contour of the superconducting microstructure.This type of magnetization distribution is not energetically favored due to large dipolar energies [30] .Consequently, the disappearance of the SC strayfield above TC is expected to lead to its relaxation to an energetically more favorable magnetic state, such as a conventional vortex or a multidomain configuration.
Here, we explore how the reduction of the lateral size of the SC structure affects the SC imprint of radial vortex-like magnetic spin textures on Ni80Fe20/YBa2Cu3O7-δ (Py/YBCO) hybrids.Finite-difference micromagnetic modelling coupled with the YBCO field modelling indicate that the radially inhomogeneous field distribution of the superconductor enables the imprint of these topologically non-trivial magnetic domain distributions below TC for lateral sizes down to sub-micrometer.
Experimentally we obtain radial vortex-like imprints down to 2 μm most likely limited by the presence of surface defects.Interestingly, although increasing the temperature above TC leads to the disappearance of the stabilizing SC stray field and the relaxation of the imprinted magnetic domain pattern, the remnant spin texture retains a significant memory of the imprinted state (non-volatile).
The robustness of this state and the origin of this memory effect is discussed in terms of pinning of domain walls by YBCO surface defects, which contribute to stabilize its topology.

Experiment
We have fabricated microstructured SC/FM hybrids with square (⊡) and disc (⨀) shape based on Fe20Ni80/YBa2Cu3O7-δ (Py/YBCO).Two types of SC/FM systems have been investigated.The first system is made of samples for which a continuous Py film has been deposited on top of YBCO structures of different sizes, as in Ref. [23] .The second type consists of samples where the Py has been structured with the same shape but with smaller sizes than that of the SC microstructure underneath (20 μm).
From now on we will use the notation (⊡, ⨀)SC Ø /FM cont to refer to (square or circular) structures with continuous Py layer and variable SC dot size, and (⊡, ⨀)SC 20 /FM Ø to refer to structures with structured Py on top of 20 μm SC dots, respectively.The superscript Ø indicates the lateral size of the SC or FM in μm as well as the size of imprinted magnetic domains.
A 250 nm thick superconducting YBCO layer was epitaxially deposited on top (001)-oriented Nb-doped SrTiO3 substrates by means of high oxygen pressure (3.4 mbar) d.c.magnetron sputtering at 900°C.Following growth, in-situ annealing in pure oxygen for 30 min at 550 º C was performed to ensure an optimal oxygen stoichiometry.Growth conditions, optimized for epitaxial c-axis growth, lead to superconducting films with transition temperature (89 K) close to that of the bulk (92 K).As-grown films are decorated by the presence of CuO surface precipitates characteristic of the high oxygen pressure sputtering growth and difficult to avoid.Precipitates are randomly distributed with diameter sizes varying between ca. 100 and 500 nm, see Figure 1 and Supplementary information of Ref. 10.
Superconducting YBCO square and disc structures, with lateral sizes (or diameter) ranging between Ø = 1 μm and Ø = 20 μm, were defined by means of electron beam lithography and etching.A 4nm thick layer of permalloy (Ni80Fe20) was deposited by means of magnetron sputtering at room temperature after the structuring of YBCO.All samples were capped with 3 nm of Al to prevent oxidation.A second lithographic process (see methods) followed in those cases where the FM was structured into the same shape as the SC.
Single out-of-plane magnetic field pulses of +100 mT or -100 mT were applied at 50 K (T < TC) to induce the supercurrent distribution that generates a magnetic stray field from the SC.Its impact on the magnetic domain structure of the FM was imaged by means of X-ray Photoemission Electron Microscopy (XPEEM).X-ray Magnetic Circular Dichroism (XMCD), measured at the Fe L3-edge (707 eV), was used as magnetic contrast mechanism (see methods).XMCD images as function of T have been obtained to evaluate the impact of the disappearance of the SC stray field above TC on the imprinted magnetic domains.All XMCD images were obtained at magnetic remanence i.e., in absence of external magnetic fields.

Results
After a zero-field cool process down to 50 K (T < TC) hybrid SC/FM structures with continuous (Figure 1a,b) and structured (Figure 1c,d) Py present a magnetic multidomain state (Figure 1e-h).An out-ofplane magnetic field pulse of -100 mT (see methods) triggers a profound modification of the Py magnetic domain distribution due to the self-field of the SC structure, see Figures 1j-l.The magnetization direction sensitivity of XMCD excludes that the resulting magnetic domain pattern is a magnetic vortex.The XMCD signal distribution for a conventional vortex would look alike to that depicted in Figures 1j-l but under a rotation of ±90 degrees (Supporting Information Section S2).The resulting magnetic domain state, both for square and disc-shaped structures, features a radial vortexlike configuration where the local magnetization is orthogonal to the microstructure contour and points towards its geometrical center [23] .Both, the resolution as well as the in-plane magnetic sensitivity of the experimental set up, prevent "visualization" of the central core predicted by micromagnetic simulations (Supporting Information Section S3).Consequently, from now on we restrict our analysis to the in-plane components of the imprinted magnetization.
In the case of discs, the radial magnetization direction features a continuous and smooth 360° rotation around its center, alike to that reported for radial vortices [9,30] .On the other hand, square structures display magnetic domain walls along the diagonals splitting the magnetic domain state into four equal triangular-shaped head-to-head magnetic domains, where the magnetization direction rotates 90° between adjacent ones.Full reversal of the magnetic domain structure can be achieved by changing the sign of the out-of-plane magnetic field pulse, whereas intermediate states can be obtained by changing its strength [23] .The effectiveness of the imprint for (⊡, ⨀)SC Ø /FM cont and (⊡, ⨀)SC 20 /FM Ø structures as function of the imprint size Ø is shown in Figures 2 and 3, respectively.XMCD images have been averaged over several similar structures to mask non-magnetic regions linked to surface defects.Both sample systems show a decrease in the efficiency of the imprint of radial vortex magnetization distributions for smaller Ø.Structures with a continuous Py layer, (⊡, ⨀)SC Ø /FM cont , show a steady decrease of the XMCD strength as the size is reduced.This can be related to the increased weight of non-magnetic regions (XMCD = 0) in the averaging and/or to a non-deterministic imprint.The latter is clear for (⊡ , ⨀)SC 2.5 /FM cont structures for which the average XMCD images (Figure 2e,f) show no traces of the SC imprint.For these samples the reduction of the lateral size of the SC dot leads to an overall decrease of the stray field of the superconductor [31,32] , see Supplementary Information Figures S1 and S2.Yet, the in-plane fields generated by (⊡, ⨀)SC 2.5 structures are high enough to align the magnetization of a 4 nm thick Py film.The efficiency of the imprint improves when the Py is structured on top of SC dots with the largest stray field (Ø = 20 μm), i.e. for (⊡, ⨀)SC 20 /FM Ø samples, see Figure 3.Such an improvement is evidenced by the fact that XMCD averaged images corresponding to (⊡, ⨀)SC 20 /FM 20 , (⊡ , ⨀)SC 20 /FM 10 , and (⊡, ⨀)SC 20 /FM 5 show a similar XMCD signal distribution, which is expected from a deterministic imprint.The imprint of domains down to Ø = 5 μm and Ø = 2 μm is possible for ⊡ and ⨀ structures, respectively, despite the large relative surface area occupied by defects (white areas in Figure 3g).
The stability of the imprint as the temperature is increased above the SC transition temperature has been investigated by obtaining XMCD images as function of temperature after a superconducting imprint at 50 K. Figure 4 depicts the data obtained for ⊡SC 20 /FM cont structures.A total of 10 similar structures have been measured.Similar results have been obtained for ⊡SC 20 /FM 20 .Inset panels a) to e) depict XMCD images obtained for one of these structures.At 80 K, close to TC (89 K), there is a partial relaxation of the imprinted magnetic domain state as evidenced by the appearance of dendritic domains and an overall change of the XMCD strength.In between 80 K and 250 K there is little variation.Similar qualitative results have been obtained for all structures measured.Fine details concerning the relaxation of the imprinted magnetic domain pattern depend on the particularities of each of the 10 systems measured, namely the distribution of defects.To wash out these particularities and reveal their common behavior we depict in inset panels f-j of Figure 4 the corresponding XMCD images averaged over all ten structures.At all temperatures, the average images feature, albeit with some relaxation at higher temperatures, the magnetic domain pattern imprinted at 50 K.Main panel of Figure 4 shows the temperature dependence of the absolute value of the averaged XMCD signal, |XMCD|, integrated over an area defined by blueish (XMCD > 0) and reddish (XMCD < 0) domains at 50 K (Figure 4f).As T increases, the |XMCD| signal decreases, reaching about 50% of its initial value at T = 90 K and remaining roughly constant up to 250 K.This reduction of the XMCD signal is associated with the decrease and final disappearance of the SC stray-field at TC leading to the relaxation of the imprinted magnetic state.The non-vanishing XMCD signal above TC indicates that the imprinted domains do not relax to a conventional vortex pattern configuration once the SC stray-field vanishes (Supporting Information Section 3), suggesting the relevance of inhomogeneous microstructure as stabilization mechanism of the imprinted magnetic configuration .

Discussion
The imprint of radial magnetic vortex configurations in samples with a continuous Py layer is not observed for confined structures with diameters below Ø ≤ 2.5 μm.This is ascribed to an interplay between the stray field generated by the superconducting layer compared to the effective anisotropy field in confined geometries, which includes the dipolar field arising from the ferromagnetic Py regions situated between the structures.Indeed, as shown in Figure 1e,f, (⊡, ⨀)SC Ø /FM cont structures exhibit a characteristic domain length comparable or larger than the lateral dimension of the largest imprinted Py region, which highlights the significant role played by the long-range dipolar field interactions originating from Py regions away from the SC structures.Structured Py samples have two benefits in this respect permitting a more efficient and a lower size-limit imprint.First, the SC stray field is maximized as they sit on top of the largest SC structures.Second, there are no FM regions in between Py structures and so no dipolar fields originating outside the SC/FM region that could compete with the SC magnetic stray field.The absence of an imprint for (⊡, ⨀)SC 20 /FM Ø structures with Ø < 2 μm indicates however, an increase of the effective anisotropy field for smaller samples.We attribute this behavior to the presence of surface defects (CuO precipitates) [10] .Indeed, micromagnetic modelling show that the stray field originating from ⨀SC 20 structures could allow the imprint of magnetic radial vortices in single crystal Py down to a lateral size of 900 nm (Supporting Information Section 3).
The presence of defects (Figure 1a-d) leads to local defective Py growth, resulting in non-magnetic regions within the FM layer, see Figure 1e-h.These defects can cause pinning and stabilization of the magnetic textures present within the spontaneous domain structure before the SC imprint.The pinning of the domain walls can lead to a reduction of the domain wall energy (local energy minimum) and to an increase of the coercivity [33] (Supporting Information Section S4).This is not surprising, since previously studies in submicrometer geometries confirmed that defects like polycrystallinity and geometrical confinement, enhance pinning and stabilization of complex topologically non-trivial textures like vortices and skyrmion tubes [34] .Consequently, the effective anisotropy field increases, and larger magnetic fields are necessary to erase the initial spontaneous domain structure.The relative importance of surface defects increases as the size of the Py dots is reduced, and so the anisotropy field eventually surpasses the available SC stray field preventing the imprint of smaller radial vortices.
Strategies to overcome the increase in coercive field due to the presence of defects would require the elimination of defects, when possible, or increasing the SC stray field.The later could be achieved either by increasing the size of the SC structures, and/or by increasing the critical current of the SC.
Worth to mention that in this case the gain in size reduction would still be limited by the size and spatial distribution of defects as well as hindered by the tendency of the magnetic ground state to evolve towards a normal vortex state [35] .
For those structures for which the SC stray field surpasses the anisotropy field, the presence of surface defects can exert a beneficial impact on the stabilization of the imprinted swirling domains across a wide temperature range.Increasing the temperature above TC leads to the disappearance of the SC stray field which stabilizes the imprint.For defect-free samples this leads to the magnetic relaxation of the system towards a normal vortex state (Supplementary information Section 3) which core polarization is determined by the z-component of the SC stray field.Conversely, the presence of surface defects can partially stabilize the imprint (Supplementary information Section 5).Indeed, the temperature dependent XMCD data shown in Figure 4 reveal that despite some relaxation, individual structures retain above TC a memory of the radial vortex imprinted state.This is evidenced in panels a) and b) of Figure 5 where the local magnetization direction averaged for (⊡, ⨀)SC 20 /FM 20 structures at 140 K (T > TC) after the low temperature SC imprint is shown by means of arrows.For both squares and discs, the angular distribution resembles that of a radial vortex despite no SC strayfield present.
Further details concerning the particularities of the SC imprint and memory effect for ⊡ and ⨀ structures can be obtained by locally comparing the angular difference between the local magnetization direction experimentally measured at 140 K (Figures 5a,b) and that expected for a radial vortex magnetic distribution.This angular deviation from the ideal imprint ( ), due to the suppression of the SC stray field, is represented in the 2D maps of Figure 5a,b by a color scale gradient.Square-like structures (Figure 5a) exhibit an asymmetric angular distribution of  with two peaks centered at ± 25° (Figure 5c).On the other hand, disc-shaped structures (Figure 5b) present a much sharper distribution of  , with a single peak centered at 0° (Figure 5d).These differences can be attributed to the presence of magnetic domain walls along the diagonals in the square structures after the low temperature imprint.These domain walls, separating head-to-head magnetic domains, increase the dipolar energy.At T > TC, in the absence of the stabilizing SC stray field, the magnetostatic energy is minimized in the vicinity of those domain by relaxing the imprinted magnetic domain state.Regions away from the diagonals, tend to be more stable (Figure 5a).In comparison, the symmetry of the discs leads to a magnetic domain imprint with no DWs as the radial magnetization rotates continuously around the center with no preferred magnetization orientation.Micromagnetic modelling confirms that the absence of DWs within the discs leads to an easier impression and to a higher stability of the imprinted state as compared to similar size square-like structures (Figure 5b and Supporting Information Figure S7).

Conclusion
Radial vortex magnetic domain configurations have been crafted in Py by means of the stray field generated by SC structures down to 2 μm lateral size.The in-plane components of this field account not only for their imprint but also for their stability at T < TC.We show experimentally that hybrid SC/FM structures retain memory of the imprinted magnetic domain state at T > TC despite the disappearance of the stabilizing SC stray field.Micromagnetic modelling indicates that instead, the stability above the superconducting transition temperature is provided by microstructural defects.IN the absence of defects, the system would relax to a conventional magnetic vortex state with its polarity determined by the z-component of the stray-field of the superconductor.
Overall, disc-shaped structures provide an enhanced preservation of the imprint than square geometries due to the due to confinement and circular symmetry, preventing the formation of headto-head 90ª magnetic domains otherwise energy-penalizing.Future work will be directed to optimize the introduction of defects (size, shape, number…) within the FM structures to improve the stabilization of the magnetic structures imprinted by the SC stray fields.Hybrid SC/FM heterostructures open an appealing direction for SC-field design and manipulation of magnetic textures in soft magnetic material, with great potential to shape spintronic applications based on magnetic textures.

Methods
Sample fabrication: Electron beam lithography was performed in a Raith50 module mounted on a Zeiss EVO 50 scanning electron microscopy instrument to obtain a square and disc pattern with different sizes.The first step was performed in a YBCO single layer using negative resist to cover parts of the layer, which was later chemically etched.A second lithography step was performed to define square and disc holes on top of the YBCO square and holes using positive resist.Py was grown on top of the sample and then a lift-off was performed to eliminate Py outside of devices.
PEEM imaging: x-ray PEEM is a magnetic and element selective technique with a resolution of ca. 30 nm.Unlike many other techniques (e.g.magnetic force microscopy), x-ray PEEM delivers direct information about the magnetization, and the element selectivity guarantees that the recorded magnetic information comes only from the element under investigation.This is important, as other techniques would also prove the magnetic field generated by the superconductor.Magnetic sensitivity arises from the difference in absorption of circularly polarized radiation with left and right helicity from a magnetic element [36] .
Experiments were done at the PEEM station at the UE49/PGMa beam line of the synchrotron radiation source BESSY II of the Helmholtz-Zentrum Berlin [37] .The angle of incidence of the incoming radiation with respect to the sample surface was of 16°, which ensured a sizable projection of the in-plane magnetization of the Py layer along the beam propagation direction, which gives rise to the XMCD signal.
Magnetic imaging was always performed in zero external field after a magnetic field pulse.The maximum pulse amplitude ± 100 mT with a pulse duration of 0.5-1 s and increasing/decreasing field rates of 10 mT/s.XMCD Images were collected at the Fe L3-edge (707 eV) for incoming circularly polarized radiation with right (σ+) and left (σ−) helicity respectively.A total of 30 images, each with a 3 s integration time, were collected per helicity.Each image was normalized to a bright field image and drift corrected before their averaging.The XMCD images were obtained as (σ- − σ+)/(σ- + σ+) where σ+ and σ− were the averaged images for right and le circular polarized radia on, respec vely.
2D maps of the magnetization direction were computed from two XMCD images obtained at 0° and 90° azimuthal rotation of the sample.XMCD images at 0° and 90° were averaged over ten similar structures.
Calculation of the stray field of the SC structures: The stray magnetic field of disc-and square-shaped superconductor structures was calculated using a 3D finite-element model based on the H-formulation of Maxwell's equation as in previous works [10,38,39] and introduced in the micromagnetic simulation as an external magnetic field.A critical current for YBCO of Jc=10 11 A/m 2 was used.
Micromagnetic simulations: Micromagnetic simulations by means of Mumax3 [40][41][42] have been performed to investigate the role of defects in the stabilization of magnetic radial vortices imprinted in Py as the temperature is raised above the superconducting transition temperature of YBCO.The ferromagnetic domain state below and above the superconducting transition temperature of YBCO was simulated by considering the presence or absence of a superconductor magnetic stray field, respectively.Ferromagnetic Py disc-and squared-shaped structures with dimension alike as those reported within the manuscript were simulated with the effective parameters of the Py layer in proximity with YBCO, calculated in previous works [43] (Msat=0.86MA/m; Aex=13pJ/m, α=3.9•10 -3 ).The presence of defects was imitated by the random inclusion of 400 holes (420 nm diameter) over an area of 20 μm X 20 μm.

Figure 1
Figure 1 XAS PEEM images obtained at T = 50 K for hybrid SC/FM structures with continuous Py layer on top a) ⨀SC Ø /FM cont and b) ⊡SC Ø /FM cont and with structured Py c) ⊡SC 20 /FM 10 and d) ⨀SC 20 /FM 10 .E-h) XMCD images obtained at T = 50 K after a zero-field cool process.I-l) Corresponding XMCD images after an out-of-plane magnetic field pulse of -100 mT.Black arrows indicate the direction of the imprinted magnetization which resembles that of a magnetic radial vortex.White arrows signal the in-coming x-ray beam direction.

Figure 2
Figure 2 XMCD images obtained at T = 50 K after a magnetic field pulse of -100 mT for a)-b) (⨀,⊡)SC 10 /FM cont , c)-d) (⨀,⊡)SC 5 /FM cont , and e)-f) (⨀,⊡)SC 2.5 /FM cont .XMCD corresponding to panels c)-f) have been averaged over 10, 7, 8 and 12 similar structures, respectively.Images corresponding to panels a)-b) correspond to a single structure.Black arrows indicate the direction of the imprinted magnetization which resembles that of a magnetic radial vortex domain.White arrows signal the in-coming x-ray beam direction.

Figure 3
Figure 3 XMCD images obtained at T = 50 K after a magnetic field pulse of -100 mT for a)-b) (⨀,⊡)SC 20 /FM 20 , and after a magnetic pulse of +100 mT for c)-d) (⨀,⊡)SC 20 /FM 10 , e-f) (⨀,⊡)SC 20 /FM 5 , and g)-h) (⨀,⊡)SC 20 /FM 2 structures.XMCD corresponding to panels a)-b), e)-g) and panel h) have been averaged over 8, 10 and 4 similar structures, respectively.Images corresponding to panels c)-d) correspond to a single structure.Black arrows indicate the direction of the imprinted magnetization which resembles that of a magnetic radial vortex domain.White arrows signal the in-coming x-ray beam direction.

Figure 4 a
Figure4a-e) and f-j) XMCD images obtained as function of temperature after a magnetic field pulse of -100 mT for ⊡SC 20 /FM cont structures.a)-e) correspond to a single structure.f)-j) are averaged over 10 similar structures.Main panel: Temperature dependence of the absolute XMCD integrated from panels a)-e) over an area defined by the "blue" and "red" triangular domains in a).A white arrow signals the in-coming x-ray beam direction.XMCD scale as in previous figures.

Figure 5 a
Figure 5 a) and b) Arrows: Angular deviation of the magnetization at 140 K after the SC imprint of a magnetic radial vortex at 50 K averaged over 10 similar (⊡, ⨀)SC 20 /FM 20 structures (see methods) in order to highlight their common behavior.Color: Angular deviation between the angular orientation at 140 K and that expected for an ideal radial vortex distribution.c) and d) histograms of the angular deviation represented in panels a) and b).Color arrows in panels c) indicate the direction of the angular deviation.