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Switchable collective pinning of flux quanta using magnetic vortex arrays

We constructed a superconducting/ferromagnetic hybrid system in which the ordering of the pinning potential landscape for flux quanta can be manipulated. Flux pinning is induced by an array of magnetic nanodots in the magnetic vortex state, and controlled by the magnetic history. This allows switching on and off the collective pinning of the flux-lattice. In addition, we observed field-induced superconductivity that originates from the annihilation of flux quanta induced by the stray fields from the magnetic vortices.


I. INTRODUCTION
The interaction between elastic lattices and fixed pinning potentials is a problem common to a variety of physical systems, e.g. repulsive colloidal particles [1] and Bose-Einstein condensates [2] in optical lattices, charge density waves in solids [3] or flux quanta (Abrikosov vortices) in type-II superconductors [4][5][6][7][8][9][10][11][12]. The phase diagram, ordering and dynamics of these systems are strongly influenced by that interaction, and ultimately by the geometry and degree of order of the pinning substrate [10,13]. This is dramatically illustrated by flux-lattice dynamics with artificial pinning potentials, where commensurability with periodic [4][5][6][7][8][9][10][11] and quasiperiodic [12] potentials induces collective or local pinning [14] and controls lattice correlation lengths. In this context, the realization of a system where the ordering of the pinning potential can be switched by an external parameter is especially interesting.
Ordered arrays of magnetic nanoparticles (dots or other geometric nanostructures) have been widely used to create pinning potentials for the flux-lattice in superconducting thin films [5][6][7][9][10][11]. In addition to the "structural pinning" (observed also in nonmagnetic structures like arrays of antidots [8]), the magnetic character of the nanoparticles generates several pinning mechanisms [9]. These include proximity effect [15], magnetic reversal losses [16], and magnetostatic interactions between flux quanta and the stray magnetic fields from the nanoparticles [17]. If the latter is the governing mechanism, the pinning potential strongly depends on the magnetic state of the nanoparticles. This gives rise to asymmetric (field polarity dependent) [6,7] flux pinning and to pinning potentials of tunable strength [11].
These effects have been observed in arrays where the individual nanoparticles present a virtually identical magnetic multidomain state and the same remanent magnetization M r .
Thus, the interaction between flux quanta and every single magnetic particle in the array is virtually identical, and changes in M r do not affect the ordering of the pinning potential, which is fixed by the array geometry.
In this article we report on a system in which the collective pinning of flux quanta can be switched on and off, as opposed to the systems mentioned above in which the ordering of the potential, and hence the occurrence of collective pinning, is fixed. This is achieved by manipulating the magnetic order of a periodic array of dipoles arranged on top of a superconducting film, as depicted schematically in Fig. 1. If the array is magnetized (all the dipoles point in the same direction), a periodic pinning potential is obtained, which causes collective pinning and induces square symmetry order in the flux-lattice. If the array is demagnetized (balanced distribution of dipoles pointing in opposite directions), a disordered pinning potential is obtained, and no commensurability develops between the flux-lattice and the array. In addition to this effect, we have observed field-induced superconductivity [18,19], which originates here from the annihilation of dipole-induced flux quanta [20]. The manipulable array of magnetic dipoles was realized using ordered arrays of magnetic dots in the so-called "magnetic vortex" state [21][22][23][24][25][26]. This system is similar to that used earlier to induce bistable superconductivity in thin films [26], except in the arrays studied here dot sizes and inter-dot distances are larger than the superconducting coherence length, giving rise to different Physics. More elaborate extensions of this experimental realization could be used to create magnetic pinning potentials of tunable geometry, asymmetry, etc.

II. SAMPLE CHARACTERIZATION
Samples consist of square arrays of Co dots ( Fig. 2  nm the dot height) was chosen [24] so that their magnetic ground-state is a "magnetic vortex" [29]. In this, the magnetization curls in-plane clockwise or counter-clockwise (vortex chirality) around a core, where it points up or down out-of-plane (vortex polarity). Fig 2 (a) shows a Lorentz image of a demagnetized array, in which the "magnetic vortex" cores appear as black (white) spots in the center of the dots for clockwise (counter-clockwise) chirality [30,31]. Further evidence of this "vortexstate" arises from in-plane hysteresis loops ( Fig. 2 (b)). These present a pronounced "pinching" in the middle of the loop, characteristic [21,25] of magnetic reversal via nucleation, displacement, and annihilation of "magnetic vortices". A cartoon of this reversal is shown in Fig. 2 (b): from negative to positive saturation (coded red (darker gray) to blue (lighter gray)), those three consecutive events are depicted. Because of the flux-closure distribution of the in-plane magnetization, the stray magnetic field from these nanodots is essentially produced by the out-of-plane magnetic moment of the vortex core, which resembles a magnetic dipole (inset in Fig. 2 (b)). As experimentally shown earlier for similar arrays of "magnetic vortices" [22,23], all of them have the same polarity in the remanent state (i.e. vortex cores throughout the array have parallel magnetization) after application and removal of a sufficiently intense out-of-plane field. On the other hand, after an out-of-plane demagnetizing cycle, the distribution of "vortex polarities" is balanced (there is an equal number of cores with magnetization pointing up/down) [22,23]. Contrary to other systems in which demagnetizing the array causes the demagnetization of each individual dot [19,32], here each dot keeps a permanent magnetic moment (the "vortex core"). Thus, these arrays of "magnetic vortices" constitute a realization of the scenario in Fig. 1. Moreover, since the insulating AlO x layer strongly reduces the proximity effect [15] between Co dots and the superconducting Al film, the magnetostatic interaction between "magnetic vortex" cores and flux quanta is the governing pinning mechanism. measurements. Therefore, in the remanent state all the vortex cores have parallel magnetization throughout the array, leading to a situation as in Fig. 1 (a). Note that this state remains unaltered during ( )

III. RESULTS AND DISCUSSION
Oe, well below the field strength needed to reverse the core magnetization (typically several kOe [22,23] Oe. These commensurability effects imply that for those fields the flux-lattice matches the square geometry of the array of "magnetic vortices" [5].
Third, is strongly asymmetric: while commensurability effects are clear at , they are barely observable at Moreover, the background resistance is larger for than for . In order to understand the behavior described above we need to consider the flux quanta induced by the external applied field, as well as those induced by the stray magnetic field from the dipoles ("magnetic vortex" cores). The total magnetic flux trough the film induced by a dipole is nearly zero [33]: as shown in the cartoon inside Fig. 2 ( [20,[35][36][37]. We will discuss later the actual situation in the studied samples. But now, let us assume that the magnetic stray field from the array of dipoles induces a certain number of flux quanta between them [38]. Fig. 4 shows a series of snapshots with the distribution of flux quanta between the dipoles as a function of the external applied field.  (Fig. 4 (d)), in which all the flux quanta are "caged" in interstitial positions by the surrounding dipoles. However, this sort of collective pinning is less effective than the one observed at , for which flux quanta are sitting directly underneath the dipoles [6,7]. This asymmetric flux pinning gives rise to the asymmetry in Fig. 3 (a). Positive flux quanta are attracted to (repelled from) dipoles pointing "up" (down). As a result, only a fraction of the flux quanta are actually pinned by the dipoles, while for others interstitial positions are more favorable. This leads to a disordered flux-lattice as well as an adverse increase in the lattice elastic energy. Because of this, collective pinning does not develop and commensurability effects are not observed (Fig. 3 (b)). [39] Finally, we discuss below the penetration of the field from the dipoles through the superconducting film, and estimate the size of the "magnetic vortex" cores. Using magnetostatics [40], we calculated the out-of-plane component of the magnetic field induced at the Al film plane by a "magnetic vortex" core. We assumed to mimic the experimental magnetization profile in a Co vortex core [30], with r the distance from the center of the core, nm [30] the radius of the core section with maximum magnetization,    Fig. 5) we estimate that the core radius nm, in good agreement with experimental values for nm for Co "magnetic vortices" [30]. 60 40 − > R 80 R As described above, we observed that the net field flux through the superconducting film underneath a dipole is . This is in contrast with previous findings for arrays of larger uniformly magnetized dots [32,34]. For these, jumps directly from to and then increases in quantized steps , with and integer [32,34]. The different behavior observed in the present experiments may be caused by the different characteristics of the magnetic field profile from the vortex cores. On the one hand, the "positive" magnetic field underneath a vortex core is highly focused (see to the non uniform magnetization within the vortex core [30] (see left inset in Fig 5), the induced field is maximum underneath its center and decreases when approaching its peripheral (see Fig. 5). Contrary to this, in the case of larger uniformly magnetized dots, the magnetic field is nearly uniform in most of the area underneath the dot and peaks near its edges [36].

IV. CONCLUSION
We have realized a superconducting/ferromagnetic hybrid system where the collective pinning of flux quanta can be switched on and off by manipulating the magnetic order of the ferromagnetic subsystem. This consists of an array of nanodots in the "magnetic vortex" state, in which crucially the nanodots have a permanent dipolar moment whose orientation can be manipulated via the magnetic history. In addition, we have observed asymmetric pinning and field-induced superconductivity effects. The latter originates from the annihilation of strayfield-induced flux quanta.

Work supported by NSF and AFOSR.
Note added. -At the time of resubmission of our manuscript we learned about the recent related publication by Hoffmann et al. [42]. In this work, the interaction of "magnetic vortices" and flux quanta is also studied. Hoffmann et al. suggest that the local suppression of superconductivity caused by the stray fields under the "magnetic vortices" is the governing flux pinning mechanism in their experiments. Contrary to this, the effects observed in our system indicate that magnetostatic interactions between "magnetic vortices" and flux quanta play a major role in flux pinning