Tunability of the critical-current density in superconductor-ferromagnet hybrids

Important modiﬁcations on the magnetization loops of the superconductor have been observed in superconductor-ferromagnet hybrids due to the effects of the ferromagnetic components, which can be used for tuning the superconductor critical-current density J c to desirable values. Here, a model based on an energy minimization procedure is presented to analyze the complex interaction between the superconductor and the ferromagnets. We show how the geometry and orientation of the ferromagnets can be chosen for shifting the position of the peaks appearing in the magnetization to positive or negative applied ﬁelds, and, consequently, to tune J c in superconductor-ferromagnet hybrids. © 2011 American Institute of Physics . (cid:4) doi:10.1063/1.3591971 The interaction between superconductors (cid:2) SCs (cid:3) and ferromagnets (cid:2) FMs (cid:3) is a very active topic of research yielding both rich physical phenomena and promising applications.


Tunability of the critical-current density in superconductor-ferromagnet hybrids
Nuria Del-Valle, Carles Navau, Alvaro Sanchez, a͒  Important modifications on the magnetization loops of the superconductor have been observed in superconductor-ferromagnet hybrids due to the effects of the ferromagnetic components, which can be used for tuning the superconductor critical-current density J c to desirable values. Here, a model based on an energy minimization procedure is presented to analyze the complex interaction between the superconductor and the ferromagnets. We show how the geometry and orientation of the ferromagnets can be chosen for shifting the position of the peaks appearing in the magnetization to positive or negative applied fields, and, consequently, to tune J c in superconductor-ferromagnet hybrids. © 2011 American Institute of Physics. ͓doi:10.1063/1.3591971͔ The interaction between superconductors ͑SCs͒ and ferromagnets ͑FMs͒ is a very active topic of research yielding both rich physical phenomena and promising applications. [1][2][3] One of the lines of research that is attracting more attention is the possibility of increasing the critical-current density J c of the SC by placing ferromagnetic components near or inside the SC. Many experiments are being carried out to study the effect of ferromagnetic parts on the J c of the SC, both in the transport case, that is, when a current is fed in the SC via an external source, [4][5][6] and in the magnetic case, in which current in the SC is induced by an external magnetic field H a . 7-10 Considering the magnetic case, it is known that in SCs the width of the hysteresis loop M͑H a ͒ is directly associated to the critical-current density J c ͑H a ͒ of the SC, 11,12 according to the critical-state model ͑CSM͒. 13 Obtaining a desired value of J c at a given field H a is, therefore, equivalent to finding a hysteresis loop with the adequate shape as to yield this J c value at H a . How to modify in general the hysteresis loop of a SC into a form corresponding to a desired value of J c ͑H a ͒? What is the effect that a given FM ͑e.g., a ferromagnetic film or a collection of magnetic dots͒ can produce in a hysteresis loop of a SC? This work aims at answering these questions. In general, the interaction of the ferromagnetic parts with the SC may occur at two different scales: at the individual vortex scale, or at that of the collective effects of many vortices. The latter situation is well described in most cases by CSM, assuming that there is a macroscopic field-dependent critical-current density; this will be the approach considered here. In this work we will present a theoretical framework and calculated results that: ͑i͒ offer a systematic understanding of the effect of a FM magnetic field in a SC M͑H a ͒ loop; ͑ii͒ explain some anomalous experimentally measured M͑H a ͒ loops; and ͑iii͒ predict that some other kind of anomalous M͑H a ͒ loop until now associated to granular materials can appear in homogeneous ͑non-granular͒ materials with a particular FM configuration.
We consider hybrid SC-FM systems composed of an infinitely long type-II superconducting strip along the x direction with rectangular cross-section a SC ϫ b SC along the y and z directions, respectively, and, as ferromagnetic part, some permanent magnets ͑PMs͒ also infinitely long in the x direction and with rectangular cross-section a PM ϫ b PM along the y and z directions, respectively. The PMs are assumed to have a constant and uniform magnetization, M PM . The system is immersed in a uniform applied field of the form H a = H a ẑ and we assume that the H a value starts at zero and changes step by step along a hysteresis loop. We consider that the SC obeys the CSM with a field-dependent critical-current density J c ͑H i ͒, where H i is the modulus of the total magnetic field ͑the sum of the applied field, the field of the PMs, and the self field created by the SC currents͒. The SC is cooled at zero applied field but with the magnetic field of the PMs. We assume that during the cooling process, no currents are induced in the SC, 14 that is, the PMs do not induce current penetration inside the SC but modify the value of J c .
The current profiles induced in the superconducting strip can be calculated by minimizing the functional given in Eq. ͑1͒ in Ref. 15, with the constraints that the net current must be zero and the current density does not exceed the J c . Introducing the J c ͑H i ͒ dependence by means of a first-order iterative algorithm, it is found that, following the notation of Prigozhin, 16 minimizing this functional with the current density J is equivalent to minimizing a functional FЈ with the current density variation ␦J defined as the difference, at present field H, between the present current density J͑r , H͒ and the previous current density Ĵ͑r , H͒ ͑at the first step of H a , Ĵ͑r , H͒ is assumed zero͒. The functional FЈ is defined as where S is the SC cross-section, A a and Â a are the vector potentials corresponding to the present applied field H a and the previous applied field Ĥ a , respectively, A ␦J is the vector potential created by ␦J, and A ⌬Ĵ is the vector potential created by ⌬Ĵ͑r , H , Ĥ ͒, defined as the increment between the previous current density at present field Ĵ͑r , H͒ and the previous current density at previous field, Ĥ , Ĵ͑r , Ĥ ͒. components of the PM-field have the vertical and/or horizontal symmetries considered in this work. Once the current densities are calculated, the magnetization of the SC, can be calculated as M = ͑1 / S͒͐ S r ϫ JdS. Due to the geometry and unless otherwise specified, the magnetization has only z-component. For this work we assume an exponential de- 17 with typical values J c0 = 4.44ϫ 10 8 A / m 2 and 0 H 0 = 5.6ϫ 10 −4 T. 7 In order to understand the effect of a PM in the hysteresis loop M͑H a ͒ of a SC we first consider, in Fig. 1͑a͒, a case of a superconducting film on top of a thin magnetic substrate. We studied the transport case for a similar hybrid SC-FM in Ref. 18 but considering the substrate as a soft FM. In Fig.  1͑a͒, we can see the effects of having a PM substrate on the M͑H a ͒ loop of a SC when the PM has a positive or negative vertical ͑z͒ magnetization, as well as the loop for the isolated SC ͑in every loop we calculate the initial curve up to a given field H a,max , the returning curve until −H a,max , and the reverse curve to H a,max again͒. 19 We observe that the peak in the returning magnetization curve occurs at H a ϳ 0 for the isolated SC ͑as expected for a thin film 12 ͒, whereas a positive ͑negative͒ magnetization in the PM substrate shifts the peak to the negative ͑positive͒ applied field values. The magnetization peaks in the reverse curve follow the same pattern. The peaks in M appear when the average field in the SC ͑the applied and self fields͒ is similar to the average of field created by the PM ͑with opposite sign͒, so that a minimum of the average of total field is produced and J c has a maximum. The described behavior is general but the details of the magnetization curve and the peak position change with varying the geometrical parameters. As an example, in Fig. 1͑b͒, we show the effect of modifying the thickness of the PM film for the case of a positive PM-magnetization ͑with negative magnetization we obtain analogous results͒ and keeping the same SC. With decreasing thickness, the field of the PM is less and compensation occurs at smaller ͉H a ͉ values, and also the height of the M peak grows until approaching its value when there is no PM. Another parameter that can be modified is the thickness of the SC. 20 Its effect on the M͑H a ͒ loop is illustrated in Fig. 1͑c͒ for the case of a SC with square crosssection on top of a thick magnetic substrate. In this case, the loop with the isolated SC shows a magnetization peak at negative H a values ͓as is known to occur for not very thin samples according to CSM ͑Ref. 12͔͒, and the effect of the PM-field yields similar effects as for films ͓Fig. 1͑a͔͒ but with wider peaks. The reason for this widening is that the SC volume is now larger and the field in its interior is more inhomogeneous so field compensation occurs more gradually.
This seems to be general behavior of the effect of a PM in the hysteresis loop of a SC: instead of having a peak at negative H a fields in the reverse curve and one at positive fields in the returning curve ͑as in standard CSM͒, both peaks are shifted to one side because of field compensations. Thus, our calculations explain the asymmetry observed in several magnetization loops of SCs with periodic arrays of magnetic dots or antidots. 1 A particularly interesting experiment performed in Ref. 7 showed a hysteresis loop of a superconducting Pb film grown on an array of Co/Pt magnetic dots, with the general characteristics described in Fig. 1. In that work, the peaks in the magnetization occurred at a positive H a field when the dots were magnetized in the positive direction ͑opposite to the results in Fig. 1͒. This can be understood from the fact that in their case of sparsely distributed magnetic dots the main contribution of the dots magnetic field to compensation is not on the areas on top of the dots ͑as considered in Fig. 1͒ but in the space between them, where the dots field has the opposite polarity. 21,22 Although the geometry of our simulated system is not exactly the same as in the experimental case ͑we consider infinitely long PMs instead of an array of magnetic dots and only consider a fragment of SC on top of two PMs͒ when taking into account a equivalent space between dots with reversed field, we show in Fig. 2  At this point we can ask whether the described behavior is the only possible effect that a PM can create on the hysteresis loop of a SC. Or its equivalent question, can J c be tuned only in this way? We will show that other interesting shapes of the loop can indeed be obtained, to be confirmed by future experiments. We now consider a PM inserted inside the SC and having horizontal ͑y͒ magnetization. In this case, the magnetization of the SC also has y-component, as shown by the antisymmetry of the current density profiles of the inset of Fig. 3 but very small ͑at most few percent͒ compared to the z-component. The calculated vertical M͑H a ͒ plotted in Fig. 3 shows an interesting shape: the peak in the reverse magnetization curve is at a positive H a and that at the returning curve is at a negative H a . This behavior has been found experimentally in cases in which the SC was granular and was associated to this granularity. 23,24 With the PM used in Fig. 3, unlike the cases of Fig. 1, the z-component of the PM-field is antisymmetric with respect to both vertical and horizontal axes, and, as the applied field is uniform, the cancellation between the PM and applied fields cannot be the reason for the peak. Instead, the z-component of the self field of the SC ͑mostly positive in the reverse curve and small͒ is the responsible that the reverse curve has a peak at positive and small applied fields, since there the sum of applied and vertical self fields compensates, on average, the z-component of the PM-field, and J c has a maximum. Thus, we theoretically predict here that such anomalous shape can arise in homogeneous SCs, so our results should be taken into account when studying granularity in SCs with FMs.
In conclusion, we have theoretically explained a set of recently published experimental data on the effect of a PM ͑or an array of dots͒ in the magnetization and critical-current of superconducting films. A continuous PM film and an array of magnetic dots may yield a different behavior, including a change in sign in the effect. Some anomalous hysteresis loops found until now only in granular system have been shown to appear also in homogeneous SCs with FMs.