phys. stat. sol. (a) 189, No. 2, 469–473 (2002)
Magnetic Shielding for Improvement
of Superconductor Performance
Y. A. Genenko1)
Institut für Materialwissenschaft, Technische Universität Darmstadt, Petersenstr. 23,
D-64287 Darmstadt, 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 Ic. 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
jc 1011 A/m2. 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
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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,
1
Hy ðx; 0Þ ¼
2p
W=2
ð
du
JðuÞ
¼ 0;
xu
ð1Þ
W=2
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
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
p ðW=2Þ2 x2
ð2Þ
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
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
pffiffiffiffiffiffiffiffi at the distance d from the strip edges
where it reaches the value of Jm ¼ I=p 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 Jb, which may be rather large, then the virgin state
of the sheet is saved unless Jm exceeds Jb. A geometrical barrier in strips of rectangular
cross-section gives [7] Jb Hc1 103 A=m, where Hc1 is the lower critical field of the
bulk superconductor, which results, for the film thickness d 102 nm, in the edge current density of the order of 106 A=cm2.
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 Jb 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 j xj 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
jnj
W=2
ð
q
du
n¼1
JðuÞ
¼ 0;
x u 2an
ð3Þ
W=2
where q is the image strength q ¼ ðm 1Þ=ðm þ 1Þ. Equation (3) may be solved exactly
in the limiting case of infinite permeability m ! 1 ðq ! 1Þ giving
JðxÞ ¼
I
cos ðpx=2aÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
a 2 cos ðpx=aÞ 2 cos ðpW=2aÞ
ð4Þ
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 ¼ a,
with a ¼ 1:01W=2. b) Current distributions in the partly flux-filled state of the strip with the total
current I=Ic ¼ 0:5; 0:7; 0:95. Regions with constant J ¼ Jc 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 Jb :
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 Ic ¼ jc 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 > Jc ¼ jc d) in the flux-free
zone. The total current in such a configuration may reach 10 Ic .
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 104 A/m, the typical
current self-field around the superconductor strips. Saturation fields of soft magnets
possessing m 104 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.
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