Magnetic Shielding for Improvement of Superconductor Performance

phys. stat. sol. (a) 189, No. 2, 469 473 (2002) Magnetic Shielding for Improvement of Superconductor Performance Y. A. Genenko 1 ) Institut für Materialwissenschaft, Technische Universität Darmstadt, Petersenstr. 23, D-64287Darmstadt, Germany (Received May 1, 2001; accepted September 30, 2001) Subject classification: 74.25.Ha; 74.60.Jg; 74.76. w; 74.80.Dm The concept of magnetic shielding for the enhancement of superconductor critical currents is discussed. Magnetic environment used for conditioning the magnetic self-field around current-carrying superconductors may drastically enhance the total critical currents of superconductor leads, resonators and electromagnets and reduce ac losses in them. Various configurations of shielding are considered and the current and field distributions are calculated to study the geometry-dependent enhancement effect. However, for successful implementation of appropriate magnet/superconductor heterostructures some special requirements on magnetic materials should be met which are discussed. Introduction The most important for applications parameter of superconductors is the maximum loss-free current they may carry, the so called critical current I c. Best critical currents achieved on practical superconductors were measured so far on thin films prepared by different techniques [1] where current densities reach the value of j c 10 11 A/m 2. This makes two-dimensional superconductors especially attractive for applications. There are two alternative loss-free states of a superconductor which carries dc transport current: flux-free, or Meissner, state when the magnetic flux is completely expelled from the sample, and a flux-filled critical state [2] when the magnetic flux penetrates (also partly) the sample but is pinned by impurities which prevents its motion and accompanying energy dissipation. In accordance to the Bean hypothesis [3] the current density in the critical state is approximately constant throughout the sample, which is to a great extent confirmed by experiment [2]. It is generally believed that an appropriate candidate for a high-current nondissipative conductor is a hard type-ii superconductor where a sufficient number of suitable effective pinning centers are introduced [1 3]. The total current in the Meissner state is normally much smaller than that in the critical state and is not considered as competitive with the latter in large-current applications. Indeed, the current flows only in a thin surface layer of a flux-free bulk superconductor or mostly along the edges of a superconductor strip resulting in a small average current. In this work, however, we show that the current distribution in a superconductor sheet may be effectively controlled by specially designed magnetic shields which enables large total current enhancements. Ferromagnetic shielding has for years been routinely used together with high-field superconductor devices like transformers or magnetic energy storage systems to prevent magnetic interference with other equipment and personnel. Recently, magnetic 1 ) Corresponding author: Tel.: +49 6151 164526; Fax +49 6151 166038; e-mail: yugenen@hrz2.hrz.tu-darmstadt.de # WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0031-8965/02/18902-0469 $ 17.50þ.50/0

470 Y. A. Genenko: Magnetic Shielding for Improvement of Superconductor Performance screening was for the first time applied actively to reduce ac losses in superconducting multifilament tapes [4, 5]. For this purpose, magnetic coating of individual filaments was implemented to suppress interfilamentary coupling that enabled ac losses reduction by more than 60 times. Magnetic materials were, however, never used to enlarge critical currents of superconductors. The Concept of Shielding of Two-Dimensional Superconductors Current and field distributions in thin superconductor sheets [6] exhibit drastic differences from those in bulk superconductors. We outline first the main features of the Meissner state in thin sheets. To calculate the magnetic field around a thin current-carrying strip one can simply sum all the contributions from single straight currents constituting the strip using Ampere s law. Then the Meissner state is defined by the condition that the field component normal to the strip vanishes everywhere at the strip surface, H y ðx; 0Þ ¼ 1 2p W=2 ð W=2 du JðuÞ x u ¼ 0 ; where the sheet current JðxÞ is the transport current density integrated over the sheet thickness d, W d is the strip width and the coordinates are shown in Fig. 1a. To the accuracy of d=w, one can consider the strip as a sheet of zero thickness which is implied in Eq. (1). The solution to this integral equation for sheet current is given by [6] JðxÞ ¼ I q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð2þ p ðw=2þ 2 x 2 where I is the total transport current carried by the strip, and shown in Fig. 1b. This current distribution is caused by the jump of the field component parallel to the strip at the strip plane y ¼ 0 because the field must detour the flux-free superconductor. It is of purely geometrical nature and does not involve any specific parameter of superconductor. The transport current (2) formally diverges at the strip edges where it draws the flux lines inside the strip and thus easily destroys the flux-free state if no barrier prevents ð1þ Fig. 1. a) The scheme of the magnetic field lines around the current-carrying superconductor strip in a flux-free state. b) Sheet current distribution over the strip cross-section

phys. stat. sol. (a) 189, No. 2 (2002) 471 Fig. 2. Field lines behavior near the edge of the isolated strip and changes in this picture made by magnetic surrounding of different forms from this. In fact, the solution (2) saturates p at the distance d from the strip edges where it reaches the value of J m ¼ I=p ffiffiffiffiffiffiffi Wd since Eq. (1) is no longer valid there. If a barrier of any nature against the flux penetration is present, it determines a critical amplitude of the edge current peak J b, which may be rather large, then the virgin state of the sheet is saved unless J m exceeds J b. A geometrical barrier in strips of rectangular cross-section gives [7] J b H c1 10 3 A=m, where H c1 is the lower critical field of the bulk superconductor, which results, for the film thickness d 10 2 nm, in the edge current density of the order of 10 6 A=cm 2. In presence of high permeability bulk magnets the magnetic field around the strip and, concomitantly, the current distribution in the strip should somehow change depending on the magnets geometry. The idea of magnetic shielding consists in attempting to control the current distribution in such a way that the total current becomes larger while the edge current peaks are kept under the critical value J b which protects the flux-free state. The possibility of this control is based on the fact that magnetic field lines are practically perpendicular to the surface of the bulk high-permeability magnets which allows one to locally manage the field lines as is shown in Fig. 2. Taking away the field lines concentration near the strip edges one may hope to reduce the edge current peaks coupled to the field. Below we present some exact and numerical solutions which quantitatively support this idea. The Straight Strip in Open Magnetic Cavities The simplest geometry of shielding which may be treated by the method of images is a strip between two magnet half spaces. Let the strip take the same position as in Fig. 1 and magnets of permeability m occupy the space jxj a > W=2 as is shown in Fig. 3a. Then the field between the magnets may be presented as superposition of the strip current self-field and of the contributions from the series of equidistant strip images [8]. Then the Meissner state equation which generalizes Eq. (1) reads n¼1 P n¼ 1 q jnj W=2 ð W=2 du JðuÞ x u 2an ¼ 0 ; ð3þ where q is the image strength q ¼ðm 1Þ=ðm þ 1Þ. Equation (3) may be solved exactly in the limiting case of infinite permeability m!1ðq! 1Þ giving JðxÞ ¼ I a cos ðpx=2aþ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð4þ 2 cos ðpx=aþ 2cos ðpw=2aþ

472 Y. A. Genenko: Magnetic Shielding for Improvement of Superconductor Performance Fig. 3. a) Superconductor strip located between two magnet half spaces with surfaces at x ¼a, with a ¼ 1:01W=2. b) Current distributions in the partly flux-filled state of the strip with the total current I=I c ¼ 0:5; 0:7; 0:95. Regions with constant J ¼ J c are flux-filled The solution exhibits homogenization and reduction of the current peaks when magnets approach the film ða! W=2Þ resulting in uniform distribution J ¼ I=W by the direct contact which leads to the filling of the whole sample with the constant current J ¼ J b : However, mostly used in applications are hard superconductors where strong pinning of the flux lines may mask edge barrier completely. In the critical state [2, 3] the current is fixed in the flux-filled domain but the flux-free part of superconductors remains very sensitive to the magnetic environment. For the above used geometry of shielding this problem may be solved exactly [9] resulting in the current distributions shown in Fig. 3b. Remarkable feature of this state is that almost the entire strip remains flux-free up to total currents slightly less than the total critical current I c ¼ j c Wd that promises low losses in ac applications. Most challenging is the realization of overcritical states in strips located in open curved magnetic cavities. An example of such cavity studied in Ref. [9] is shown in Fig. 4a. Conformal mapping of this geometry to that of Fig. 3a enabled calculation of Fig. 4. a) Shielding configuration obtained by conformal mapping of that in Fig. 3a with the distance a ¼ 1:04W=2 fixed. b) Current distributions are plotted for different values of a parameter b presenting the flux-free fraction of the strip

phys. stat. sol. (a) 189, No. 2 (2002) 473 the distributions plotted in Fig. 4b which are overcritical (J > J c ¼ j c d) in the flux-free zone. The total current in such a configuration may reach 10 I c. Discussion:Appropriate Magnetic Materials Needed An important feature of the magnetic shielding established in numerical studies [9, 10] is that the changes in current distributions saturate already at moderate values of permeability so that 99% of effect is reached by m 200: This makes analytical solutions obtained in the limit m!1[8, 9] representative with high accuracy for magnets with m of some hundreds. Nevertheless, much higher m would be of benefit to the effect in real experimental conditions. Indeed, all the above results were calculated for infinitely extended shields but the field in the two-dimensional geometry decreases with growing distance very slowly so that finite size of bulk magnets may affect the experiment. Besides, magnets should be very soft at the typical temperature of 77 K since any remanent magnetization over 1 mt would negatively affect superconductors. Maybe the most crucial problem is a saturation field of magnets which must be as large as 10 4 A/m, the typical current self-field around the superconductor strips. Saturation fields of soft magnets possessing m 10 4 are usually much smaller, therefore some compromise is desirable to obtain cryogenic materials with larger saturation fields and still high enough m. Also desirable would be the low magnetic viscosity (short relaxation time) reducing magnetic losses in ac applications. References [1] H.-W. Neumüller, W. Schmidt, H. Kinder, H. C. Freyhardt, D. Stritzker, R. Wördenweber, and V. Kirchhoff, J. Alloys Compd. 251, 366 (1997). [2] A. M. Campbell and J. E. Evetts, Critical Currents in Superconductors, Taylor & Francis, London 1972. [3] C. P. Bean, Phys. Rev. Lett. 8, 250 (1962). [4] M. Majoros, B. A. Glowacki, and A. M. Campbell, Physica C 334, 129 (2000). [5] B.A. Glowacki and M. Majoros, Supercond. Sci. Technol. 13, 483 (2000). [6] E. H. Rhoderick and E. M. Wilson, Nature 194, 1167(1962). [7] M. V. Indenbom, H. Kronmüller, T. W. Li, P. H. Kes, and A. A. Menovsky, Physica C 222, 203 (1994). [8] Yu. A. Genenko, A. Usoskin, and H. C. Freyhardt, Phys. Rev. Lett. 83, 3045 (1999). [9] Yu. A. Genenko, A. Snezhko, and H. C. Freyhardt, Phys. Rev. B 62, 3453 (2000). [10] Yu. A. Genenko, A. Usoskin, A. Snezhko, and H. C. Freyhardt, Physica C 341, 1063 (2000).