Superconducting imprint of magnetic textures in ferromagnets with perpendicular magnetic anisotropy

Research on proximity effects in superconductor/ferromagnetic hybrids has most often focused on how superconducting properties are affected—and can be controlled—by the effects of the ferromagnet’s exchange or magnetic fringe fields. The opposite, namely the possibility to craft, tailor and stabilize the magnetic texture in a ferromagnet by exploiting superconducting effects, has been more seldom explored. Here we show that the magnetic flux trapped in high-temperature superconducting YBa2Cu3O7-δ microstructures can be used to modify the magnetic reversal of a hard ferromagnet—a cobalt/platinum multilayer with perpendicular magnetic anisotropy—and to imprint unusual magnetic domain distributions in a controlled manner via the magnetic field history. The domain distributions imprinted in the superconducting state remain stable, in absence of an external magnetic field, even after increasing the temperature well above the superconducting critical temperature, at variance to what has been observed for soft ferromagnets with in-plane magnetic anisotropy. This opens the possibility of having non-trivial magnetic configuration textures at room temperature after being tailored below the superconducting transition temperature. The observed effects are well explained by micromagnetic simulations that demonstrate the role played by the magnetic field from the superconductor on the nucleation, propagation, and stabilization of magnetic domains.


Characterization of hybrid SC/FM structures
The as-grown YBCO film is decorated, as typically found in thin YBCO films, by the presence of CuO precipitates spaced several microns (Fig. 1c of main manuscript). The YBCO grows around screw dislocations which for the thickness range of these experiments give rise to micron size square pyramids with flat and continuous terraces, see Figure S1a. The deposition of a FM layer by means of sputtering covers well these terraces with flat and non-granular surfaces ( Figure S1b) confirmed by the presence of large magnetic domains (Fig. 1d within manuscript). XMCD-PEEM images of the domain structure of Co/Pt grown on YBCO (Fig. 1d) are very similar to that measured by magnetic force microscopy for Co/Pt grown o Silicon wafer. This demonstrates that the choice substrate (YBCO or Si) have no effect on the magnetic properties, which are the standard, well documented ones for Co/Pt multilayers. The effect of the CuO precipitates and the patterning on the coercivity, residual magnetization and magnetic domain structure, has been investigated by analyzing the full-field images of the XMCD-PEEM data depicted in Figure 2 for T > Tc. Full-field images are shown in Figure S2. The magnetic domain structure of (Co/Pt) is similar, within resolution, independently on whether the FM is deposited on top of the SC dot (central square) or directly on top of the substrate (panels a-e). The residual magnetization is almost 100% in both regions (panel f) as expected for systems with perpendicular magnetic anisotropy. The "averaged" switching field for the FM on top of the SC is lower as compared to that on top of the substrate. This is likely to be due to the presence of CuO segregation on top of the YBCO square which acts as nucleation sites for reversed-magnetization domains.

Magnetic field generated by a superconductor square
The magnetic field generated by the superconductor square ( ⃗ ⃗ sc ) during and after the magnetic field application has been obtained by a 3D finite-element method based on the H-formulation of Maxwell's equations implemented in the finite-element software package Comsol Multiphysics 2,3 . The model solves Faraday's equation using a nonlinear resistivity for the superconductor material 4 where Ec is a threshold electric field, Jc is the critical current of the superconductor, and n a power index describing the steepness of the superconducting transition. Eq. (1) approximates Bean's critical state model 5 for sufficiently large values of n. For the simulations used for this work, the following values were used: Ec=10 -4 V/m, J c =10 11 A/m^2 and n=25.
Simulations have been done for a squared superconductor structure with dimensions akin to those of the real system, i.e. 13 µm X 13 µm X 130 nm. The superconducting parallelepiped is surrounded by a cubic air domain. The external magnetic field is applied as a Dirichlet boundary condition on the face of the cube along the z axis. On the symmetry planes parallel and perpendicular to the external field, conditions of magnetic insulation and perfect magnetic conduction are respectively applied. The air is modeled as a material with very large electrical resistivity (1 Ωm). The superconducting and air domains are meshed with hexahedral and tetrahedral elements, respectively. In particular, the superconducting domain is meshed with a grid of 24x24x3 elements (See Figure S3).
The magnetic field history followed for the simulations was the following; after a pulse of

Figure S4
Magnetic field history followed for the calculations of the superconducting magnetic field components. Open dots at pulse peak and filled dots at pulse end. Inset shows detail for time up to 0.020 sec.
Error! Reference source not found. and Error! Reference source not found. show 2D maps of the computed y component of the supercurrent (J) and x and z components of H ⃗⃗ sc , respectively. J x (x,y) and H y sc (x,y) can be obtained by a rotation of 90° of J y (x,y) and H x sc (x,y). Top and bottom rows correspond to maps obtained at the peak of the magnetic field pulse (open dots in Figure S4) and at pulse end (filled dots in Figure S4). At remanence, both the components of the supercurrent and the superconducting field change with H z pulse strength. While the supercurrent (Fig. S5) seems to saturate at about 40 mT, the Z-component of H z sc (x,y) still changes with H z pulse . The latter is clearly seen at the center of the square where we see that H z sc (x,y) is different at 'pulse end' (filled dots) for µ 0 H z pulse = 40 mT, 44 mT and 100 mT, panels j, k and l, respectively.
The magnetization of a SC structure vs an external out-of-plane magnetic field follows a characteristic hysteretic loop 6 . Full remanence of the magnetization within the SC is achieved for out-ofplane external fields H z ext > 2H p where H p is the so called full penetration field 6,7 . Unless explicitly mentioned, from now on we will discuss the supercurrent and superconducting field components at    Figure S8 shows the integration of the normalized y component of the supercurrent for x < 0 as function of H z pulse at 'pulse end' (squares). As comparison we also plot H z sc (0,0) at the center of the square (dots). While the current saturates at µ 0 H z pulse ≈ 24-28 mT, the z-component of the superconducting field at remanence (filled dots) does not saturate until ca. 52 mT. We deduce that µ 0 H p is ca. 26 mT We note that the field pulse range for which our micromagnetic simulations show a position dependent switching field for the Co/Pt multilayer is between 40 and 50 mT (Fig. 4b main text and orange bar in Figure S8). This field range stays within the field region for which ⃗ ⃗ sc of the superconductor is not saturated i.e. H z s < 2H p . Hence, switching of the magnetization orientation for the Co/Pt multilayer takes place in a region where the superconducting field components do change with H z pulse (Fig. 4 within main text and Figs. S4 and S5). Full switching is achieved at field (ca. 50 mT) very close to that for which the superconducting field saturates.
For H z ext > 2H p the superconducting stray fields for the external field applied ('pulse peak') and at remanence ('pulse end') do not change and are the same except for a different sign. This can be seen for µ 0 H z pulse = +100 mT in Figure 4f    Among these parameters, we measured magnetization loops (parallel and perpendicular to sample plane) in twin samples using alternating gradient force magnetometry (AGFM) or SQUID and we evaluated the magnetization at saturation, M s , and the effective out-of-plane anisotropy, K eff . For the simulation, we hence considered the following measured Co properties: M s = 1.6 MA/m (this value is large due to the proximity effect in Pt), the effective interfacial anisotropy K u = 1.8 MJ/m 3 , and the exchange A = 12 pJ/m. A is not directly measured in this study, but is estimated from previous studies 9 .
To keep a manageable numerical problem (memory wise), we considered rather large cells (3 nm in plane, 8 nm out-of-plane), using a single effective magnetic layer. The parameters are adapted using an effective medium approximation as explained in, e.g., Ref. 10 , with the effective parameters (denoted by , with f = 0.6/1.6 = 0.375 the ratio of the Co thickness over the total (Co|Pt) thickness. Because in ideal simulation the symmetry must be broken to allow nucleation (no thermal fluctuation is considered) and because small misalignment of the actual sample in the magnetic field is unavoidable, we considered a 0.5° angle between the applied field and the normal of the magnetic film (if not indicated otherwise). We know that our films, grown by sputtering at room temperature, are (1 1 1) textured with grains of typical lateral size of about 10 nm. This structure generates pinning that is modeled here as a 9% reduction of the exchange between grains, and 9% variation (standard variation) of the M s , e.g. mimicking thickness variation 11 . This grain structure is necessary to stabilize the intermediate magnetization states that are observed. The behavior of the magnetic film is calculated by sequential minimization of the energy 12 using the external fields given by the finite element calculation of the fields generated by superconducting currents (see previous paragraph). The misalignment is simply modeled by an in-plane component of sin(0.5°) H z ext .
The micromagnetic calculations indicate that the reversal of the magnetization occurs through nucleation and propagation of the magnetic domains. Nucleation mostly occurs at edges and is favored by in-plane fields. In Figure S9, the displayed switching fields H z as a function of in-plane field H x demonstrates that H z is reducing by much more than H x . (For these micromagnetic simulations we used the same 8-nm-thick film, but we reduced its size to 1.3 x 1.3 µm².) On the other hand, the propagation depends on the external out-of-plane field. With the chosen disorder landscape (grain structure) and misalignment of the field, the domain propagation starts above about 20 mT. One can note that the metastable states obtained during the magnetization reversal display a magnetization texture with small (few hundred nm wide) meandering domains, so-called labyrinthine or worm domains, with up/down domain ratio depending on the field history. In this system, a smooth micrometer long domain wall would be destabilized by the large dipolar energies, and the system reduces the total magnetic energy by forming these worm domains.

Effect of full saturation field on switching field profiles
The 3D finite-element calculations show that the field pulse range for which our micromagnetic simulations show a position dependent switching field for the Co/Pt multilayer stays within the field region for which ⃗ ⃗ sc of the superconductor is not saturated i.e. H z s < 2H p , see Figure S8. In the following we address how the local switching fields depend on the relation between the magnetic field pulse range and the full penetration field (H p ) of the SC.
We considered three different scenarios: i) "Real field" case (blue curve): Micromagnetic simulations are fed with the fields of the superconductor obtained after following the magnetic field history depicted in Figure S4.

Figure S10
Line profiles of the switching field obtained by finite elements and micromagnetic simulations for the three cases described within the main text. Figure S10 show the line profiles obtained for the switching field in the aforementioned cases, they all show a switching field which value depends on position. The line profile of the switching field obtained for the "high field" case (green) shows an effect smaller and with a smoother profile than that experimentally observed. The line profile obtained for the "low field" case shows the largest variation in switching field between the edges and the center of the structure and is the one which better resembles our experimental data. We note, however, that this case does not reflect the physics of our experiment; as an initial state we have assumed a saturated state for the magnetization of the SC, we have also assumed that H pulse << 2H p . The later means that the H pulse range is not strong enough to modified the initial state. Experimentally this is contradictory; if the H pulse range used is not able to modify the initial saturated state, the same H p range would have not been enough to set this initial saturated state. The switching field profile for the "real" case is an intermediate case which reproduces fairly well our experimental observation that is, a switching field which varies faster close to the center of the structure than close to the edges.
The differences observed in the line profiles of the switching field for the 3 scenarios can be understand by considering their different H z sc profiles ( Figure S11). As pointed out within the manuscript our initial simulations pointed out that "… the domain propagation is locally activated when the magnetic field µ 0 H z (x,y) is above a threshold µ 0 H z th ≈ 20 mT". Figure S11 shows the line profiles of H z sc at pulse peak (dotted line) and at pulse end (continuous line) for the "high" and "low" field cases.
Nucleation of reversed domains should take place at similar values for both "high" and "low" field cases as the z-profile of the stray field of the superconductor at the peak of the magnetic pulse is the same (dotted curves in Figure S11), see manuscript for details. After nucleation, domain propagation takes place. A reversal of the magnetization for the full structure is achieved once the z-component of the effective field is larger than the threshold field µ 0 H z th at every position (dashed line in Figure S11). For the "low field" case the field pulse necessary to fulfill this condition is ca. 57 mT. This value is the one necessary so that the center of the structure (µ 0 H z sc ≈ -37mT) reaches an effective field of 20 mT during the peak of the pulse (vertical blue line in left panel of Figure S11). For the high field case, a smaller field (ca. 40 mT) is enough to fully reverse the magnetization direction of the square structure. This is due to the fact that the central region of the square at pulse end always fulfills µ 0 H z sc > µ 0 H z t . A pulse of 40 mT is enough to bring above the threshold the effective field for the rest of the square during the pulse peak (vertical blue line in right panel of Figure S11).
These finite element and micromagnetic simulations strength the conclusions reported within the main manuscript and in addition highlight the importance of the relative size between H pulse and H p .

Figure S11
Line profiles of the z-component of the stray field of the superconductor at pulse peak (open dots) and pulse end (continuous line) for the "high" (H pulse > 2H P ) and "low" (H pulse << H P ) field scenarios, see text. The vertical blue line indicates the external field necessary so that the threshold field for domain propagation is fulfilled for the complete FM square.

Effect of pulse sweep rate and flux relaxation on imprinted magnetization distributions
The effect of the magnetic field pulse sweep rate on the imprinted magnetization distribution has been investigated by finite element calculations of the stray field of the superconductor. As argued within the main text, the z-component of the stray field of the superconductor (H z sc (x,y)) at pulse peak is a key parameter to determine the extension of the switching. Consequently we have investigated how the z-component of the effective field, H z (x,y) = H z pulse + H z sc (x,y), changes depending on the switching field rate. Calculations have been restricted to 2D for the sake of keeping them within a reasonable time. Calculations have been done for infinitely thin disc of 13 µm diameter and for µ 0 H pulse ≈ 65 mT (conditions alike to those reported in Figure 5 within main manuscript). Figure S12 show radial plots of H z profiles for different sweep rates at peak (left panel) and end (right panel) of the pulse. We observe that as the sweep rate for the magnetic pulse decreases: i) At pulse peak (left panel): a. The effective field at the edges of the SC decreases slightly.
b. The effective field at the center of the structure increases.
ii) At pulse end (right panel): a. The effective field at the center of the structure decreases slightly.
These observations leads to the following conclusion: decreasing sweep rates will require slightly larger fields for the nucleation of inversed magnetization domains and slightly smaller field pulses to reach full switching. The tendency shown by this "fast" simulations make us confident that these results can be reasonably extended to slower sweep rates (experiment sweep rate was 0.001Ts -1 ).
While it is true that the sweep rate have an effect on the nucleation and propagation of inversedmagnetizatio domains our results indicate that this effect is small, i.e. minor modification of the fields required for nucleation and full propagation.
In addition, magnetic relaxation of the SC remnant state will lead to a weakening of the effective field and thus will not modify the imprinted magnetic state.

Figure S12
Radial plot of H z at peak pulse (left) and pulse end (right) as function of pulse sweep rate.
The propagation of the inversed-magnetization domain towards the center takes mostly place at peak of the pulse as the z-component of the effective field reaches its maximum. Magnetic relaxation of the SC remnant state 13 will lead to a weakening of the effective field and thus will not modify the magnetic imprint.

PEEM on hybrid SC/FM circle structures
Figures S13, S14 and S15 show the XMCD images obtained for 15 hybrid structures with circular shape (diameter 13 µm).
A series of XMCD images as function of H z pulse were obtained for structure 1a ( Figure S13 and  Figure S13-Figure S15 show the XMCD images obtained at 50 K for the 15 structures after the sample followed the determined magnetic field history (see main text). Images at 120 K (T > T sc ) where obtained after 30 minutes at the target temperature (middle of images on Figs. S13-S15). The magnetic domain pattern imprinted at 50 K is still clearly visible without appreciable changes at 120 K, small changes in XMCD contrast are attributed to differences in focusing conditions and beam position and Figure S13 XMCD images obtained for hybrid SC/FM structures 1a to 1e after superconducting magnetic field imprint at 50 K (top row), after increasing T to 120 K (middle row) and after 20 min. at 300 K and cooling back to 120 k (bottom row).
beam drift during data acquisition. Increasing the temperature to room temperature leads to degassing of the sample and an increase of the pressure at the PEEM which preclude imaging at this temperature.
We therefore obtained images at 120 K after keeping the sample at RT for 20 minutes. The obtained images are shown in the bottom row of images in Figs. S13 to S15. The magnetic domain texture does still resemble the one imprinted at 50 K.
Figure S15 XMCD images obtained for hybrid SC/FM structures 3a to 3e after superconducting magnetic field imprint at 50 K (top row), after increasing T to 120 K (middle row) and after 20 min. at 300 K and cooling back to 120 k (bottom row).

Figure S14
XMCD images obtained for hybrid SC/FM structures 2a to 2e after superconducting magnetic field imprint at 50 K (top row), after increasing T to 120 K (middle row) and after 20 min. at 300 K and cooling back to 120 k (bottom row).