Supercond. Sci. Technol. 9 (1996) 1042–1047. Printed in the UK
Penetration dynamics of a magnetic
field pulse into high-Tc
superconductors
V Meerovich†, M Sinder‡, V Sokolovsky†, S Goren†, G Jung†,
G E Shter§ and G S Grader§
† Physics Department, Ben-Gurion University of the Negev, POB 653, 84105
Beer-Sheva, Israel
‡ Material Engineering Department, Ben-Gurion University of the Negev, POB 653,
84105 Beer-Sheva, Israel
§ Chemical Engineering Department, Technion, 32000 Haifa, Israel
Received 30 May 1996, in final form 28 August 1996
Abstract. The penetration of a magnetic field pulse into a high-Tc superconducting
plate is investigated experimentally and theoretically. It follows from our
experiments that the threshold of penetration increases with increasing amplitude
and/or decreasing duration of the applied pulse. The penetrating field continues to
grow as the applied magnetic field decreases. The peculiarities observed are
explained in the framework of the extended critical state model. It appears that the
deviations from Bean’s classical critical state model are characterized by a
parameter equal to the square of the ratio of plate thickness to skin depth. The
applicability of the classical critical state model is restricted by the condition that
this parameter is much less than 1. This condition is also the criterion for the
applicability of pulse methods of critical current measurements.
1. Introduction
Bean’s critical state model (CSM) [1] is used to describe
electromagnetic properties of low-Tc superconductors up to
frequencies of about 10 MHz. Investigations [2, 3] showed
that this model also describes the transport and magnetic
properties of bulk ceramic high-Tc superconductors
(HTSCs). However, for HTSCs, deviations from this
model were found at much lower frequencies. The authors
of [4] observed a frequency dependence of the shielding
factor; in [5] the surface impedance was found to be
frequency dependent; in [6, 7] it was shown that the loss
per cycle depends on frequency and deviates from the
cubic dependence on the magnetic field. Because these
peculiarities cannot be explained in the framework of the
CSM, some nonstationary models have been suggested
[6–9]. These models take into account the motion of fluxons
and its influence on the distribution of magnetic field in the
superconductor. In fact, the magnetic behaviour of HTSCs
under a time-varying field is influenced by the associated
resistive dissipation. To take into account this peculiarity,
Bean has proposed an extension of the CSM that includes
viscous motion of the fluxons [9]. This extended CSM
was fruitfully used to investigate the frequency dependence
of AC losses [6]. The second important problem is the
investigation of the penetration of time-varying magnetic
fields into superconductors. This investigation is essential
both for development of superconducting shields and
c 1996 IOP Publishing Ltd
0953-2048/96/121042+06$19.50 for analysis of the results of measurements using the
pulsed technique [10–13]. In [11–14], critical current
measurements in pulsed magnetic fields were reported. The
analysis of the penetration of a pulsed magnetic field into
a superconductor was based on the CSM [14, 15].
Bean used the extended CSM to study the response of
an HTSC to a step in the magnetic field [9]. He found that
the magnetic field distribution tends with time to that given
by the classical CSM. In our previous paper [16], we used
the extended CSM to investigate theoretically the magnetic
field penetration into a semi-infinite HTSC slab when an
external magnetic field was increased monotonically with
time according to a power law. We obtained the result that,
for high rates of charge, of an applied magnetic field, the
solution does not tend to that given by the classical CSM
and the penetration depth is substantially less than the depth
predicted by this model. Developing these investigations,
it is natural to consider the penetration dynamics under
application of single pulses. This paper presents the results
of an experimental and theoretical study of the response of
HTSCs under an applied magnetic field pulse.
2. Experimental procedure
The experimental study was based on measurements of
the temporal dependence of the magnetic field penetrating
through an HTSC plate exposed from one side to a magnetic
Penetration of magnetic field pulse into HTSC
Figure 1. Experimental arrangement for measurements of
penetration of magnetic field.
field pulse. The schematics of our experimental set-up is
shown in figure 1. A driving coil and a Hall probe for
measuring the magnetic field were placed on the opposite
sides of a superconducting sample. The cylindrical driving
coil was 14 mm in diameter and 24 mm in length and
contained 600 turns of 0.3 mm diameter copper wire. The
coil together with a superconducting sample were cooled
to 77 K by liquid nitrogen. The Hall probe had an
active area of 0.03 mm2 and was arranged to measure the
component of the local penetrating magnetic field parallel
to the applied field. Values of the current in the driving
coil and signals from a Bell Gaussmeter (model 9200
with sensitivity 0.01 G) were recorded by a two-channel
digitizing oscilloscope and passed to a computer for storage
and analysis.
A single current pulse of duration about 2.5 ms and
of controlled amplitude up to 250 A was obtained by the
discharge of a capacitor. Short times of high current pulses
allow one to neglect heating processes.
Two superconducting samples prepared by various
technologies from YBCO and BSCCO were used in the
present study. The BSCCO 2212 sample shaped as a plate
of size 110 mm×95 mm was cut from a cylinder 200 mm in
diameter, 95 mm in length and 3.5–4 mm in wall thickness.
The cylinder was fabricated by centrifugal casting of a
homogeneous melt from the oxides of Bi, Sr, Ca and Cu
with an addition of 10% SrSO4 [17]. The YBCO 123 discshaped sample was 32 mm in diameter and 2.5 mm in
thickness, prepared by conventional sintering of oxalatederived powder [18].
Figure 2. Scope traces of driving current and penetrating
magnetic field for the BSCCO 2212 plate-shaped sample.
3. Experimental results
• leakage field around the edges of the superconducting
samples;
• microcracks or defects in the samples [19];
• flux motion viscosity [20];
• penetration and trapping of the magnetic field in
intergranular space [21];
• inhomogeneity of the critical current density which
leads to a change in the pattern of shielding current paths
and to partial field penetration [22].
Figure 2 illustrates a typical result of the penetration
experiment: the temporal dependences of the current in the
driving coil and corresponding signal from the magnetic
probe located on the opposite side of the BSCCO sample.
The coil axis was directed parallel to the sample surface.
The penetrating magnetic field as a function of the
instantaneous value of the current in the pulse is presented
in figure 3. The applied magnetic field is proportional to
the current in the driving coil: a driving current of 1 A
Figure 3. Penetrating magnetic field versus driving instant
current for various amplitudes Imax of a pulse
(BSCCO 2212 sample in parallel field).
corresponds to a magnetic field of 5.4 G at the point of the
probe location at a temperature of 300 K.
Penetration of the magnetic field through the plate exists
already at applied fields much lower than the threshold of
flux breakthrough. This peculiarity is observed by many
authors in magnetic shielding experiments [19–22] and
explained by the following:
Because of the smooth change of penetrated field with the
1043
V Meerovich et al
Figure 4. Penetrating magnetic field versus driving instant
current for various amplitudes of alternating current
(BSCCO 2212 sample in parallel field).
Figure 5. Penetrating magnetic field versus driving instant
current for pulse and direct currents (YBCO 123 sample in
perpendicular field).
current, it is difficult to determine exactly the value of the
driving current at which the breakthrough occurs. However,
as can be seen from figure 3, the sharpest increase of
the penetrating field is observed when the current is close
to its maximum. The rate of increase of the penetrating
field is higher than that of the applied field. As the
amplitude of the pulse increases, the field breakthrough
starts at a higher instantaneous current. The penetrating
field continues to grow as the current and, hence, applied
magnetic field decrease. Moreover, a certain increase of the
penetrating field is observed after the applied pulse is over.
Analogous but less pronounced peculiarities were observed
when the sample was affected by a 50 Hz AC magnetic
field (figure 4).
We have found that the penetration peculiarities were
preserved with a change of the direction of the coil axis
and material of the samples. For example, the penetration
of the magnetic field into a YBCO tablet when the coil axis
is perpendicular to the sample surface is shown in figure 5.
For comparison, the curve representing the penetration of a
DC magnetic field is also given in this figure. The magnetic
probe was arranged to measure the field component normal
to the sample surface.
From figure 5, the breakthrough of the DC magnetic
field becomes appreciable at a current of about 0.35 A. For
pulse fields the breakthrough current is substantially higher
and increases with an increase of the pulse amplitude,
attaining 0.45 A for Imax = 0.95 A and 0.65 A for
Imax = 2 A.
simplified case of an infinite superconducting plate exposed
to a pulsed magnetic field applied parallel to the plate
surface, in the z direction. The positive x direction is
chosen inward and perpendicular to the plate, and currents
are induced along the y axis.
Maxwell’s equations specialized to our geometry are
4. Theoretical analysis
The observed non-linear peculiarities of the penetration
of a time-varying magnetic field into HTSCs cannot be
explained in the framework of the classical CSM. We will
use an approach of the extended CSM which is based
on Maxwell’s equations supplemented by a constitutive
relationship between an intensity of an electric field E
and a current density j . We will restrict ourselves to the
1044
∂E/∂x = −µ0 ∂H /∂t
(1)
∂H /∂x = −j
(2)
where H is the intensity of the magnetic field. Following
Bean [9], we use the relationship between E and j in the
form
E = ρf (j − jc ), E > 0;
E = ρf (j + jc ), E < 0 (3)
where ρf is the flux flow resistivity and jc is the critical
current density.
To determine the parameters ρf and jc , several samples
were cut out from the BSCCO sample and tested by a fourpoint pulse method described in [10]. The samples had
weak sections with a cross-section of about 2 mm×1.5 mm
and a length of 5 mm. The current pulse was of the same
shape and duration as that used in our magnetic penetration
experiment. The different samples gave values of the
critical current density in the range 1000–1400 A cm−2 and
the flux flow resistivity in the range 8×10−9 –3×10−8  m.
The same procedure applied to the YBCO sample gave
about 400 A cm−2 and 10−8  m, respectively. The
critical current of the samples decreased by about 20% in
a magnetic field of 200 G [17], which was the maximum
magnetic field in our experiments.
Equations (1) and (2) are rewritten in dimensionless
units:
∂ Ẽ/∂ x̃ = −a∂ H̃ /∂ t˜
(4)
∂ H̃ /∂ x̃ = −j˜
(5)
where Ẽ = E/ρf jc , H̃ = H /1jc , j˜ = j/jc , a =
µ0 12 /tx ρf , x̃ = x/1, t˜ = t/tx , tx is the characteristic time
Penetration of magnetic field pulse into HTSC
of the change of the applied field and 1 is the thickness
of a superconducting plate. The critical current density is
assumed to be constant. For a pulse field, tx is the pulse
duration equal to 2.5 ms in our experiments and hence
a = 0.2–1. For an AC magnetic field, tx can be chosen
as 1/ω, where ω = 2πf ; f is the frequency of the applied
magnetic field. In the last case a = 12 /2δ 2 , where δ is
the skin depth determined as for a normal metal with the
resistivity equal to ρf .
The relationship (3) in dimensionless form is
Ẽ = j˜ + 1, Ẽ < 0.
Ẽ = j˜ − 1, Ẽ > 0;
The boundary and initial conditions in the dimensionless
units are
H̃ (t˜ = 0, x̃) = 0
H̃ (t˜, x̃ = 0) = H̃0 F (t˜)
Ẽ(t˜ = 0, x̃) = 0
(6)
H̃ (t˜, x̃ = x̃p ) = 0
Ẽ(t˜, x̃p ) = 0
(7)
where H̃0 = H0 /jc 1, H0 is the amplitude of the magnetic
pulse, F (t˜) is the law of the change of the applied magnetic
field with time and x̃p is the penetration depth of the
magnetic field. The last two boundary conditions in (7)
are used when the penetration depth is less than the plate
thickness. When the magnetic field penetrates through the
plate, the last two boundary conditions (7) are changed
by the integral energy conservation law in the plate to the
form
Z 1 2 Z 1
∂
H̃
Ẽ j˜ dx̃ + a
dx̃
(8)
Ẽ1 H̃1 = Ẽ2 H̃2 +
˜
2
∂
t
0
0
where Ẽ1 , H̃1 , Ẽ2 and H̃2 are the intensities of
electric and magnetic fields on both surfaces of the
plate.
The boundary conditions (7) are valid as long as
Ẽ1 = Ẽ(x̃ = 0) > 0. When an applied magnetic field
decreases, the electric field also decreases and can reverse
its sign on the surface. From this time on, two zones with
opposite signs of Ẽ appear just as in the classical CSM.
The conditions on the boundary of these zones are
• continuation of the magnetic field,
• the current density jumps from −1 to +1 when
passing through the boundary.
Equations (4) and (5) with the corresponding initial and
boundary conditions were solved numerically. The results
are presented in figures 6–9.
Figure 6 reflects the
penetration process for different values H̃0 and a.
The penetration depth is determined by the condition
that the induced current shields fully the applied magnetic
field. For the case xp ≤ 1, we have
Z xp
j (x) dx.
H (t, x = 0) =
0
In the classical CSM, j = jc and the penetration depth
increases proportionally with applied magnetic field. The
maximum depth is equal to H0 /jc . In the extended
model, the induced current exceeds considerably the critical
Figure 6. Theoretical temporal dependences of penetration
depth: (a ) for various values of the parameter a at H̃0 = 1;
(b ) for various amplitudes of an applied magnetic field
pulse H̃0 at a = 1. xm is the maximum penetration depth in
the classical CSM.
value and limits the penetration of magnetic field into a
superconductor (figure 7). The distribution of an addition to
the critical current in principle is the same as the distribution
of induced currents in normal metals. The difference is
that the current falls sharply to zero at the point x̃p . As in
normal metals, there is a temporal shift between the applied
magnetic field and the induced current. So, at t˜ = 0.38,
when the applied magnetic field achieves its maximum
value the current density near the plate surface is about
1.5 times greater than the critical one. The penetration
depth decreases with growth of the parameter a and can
be sufficiently lower than the Bean depth determined by
the classical CSM (figure 6(a)). So, at H̃0 = 3 and
a = 1, the depth is only one-half of Bean’s penetration
depth. In the classical CSM, a zone with the opposite sign
of the current appears exactly at the point when the time
derivative of an applied magnetic field reverses its sign.
As distinct from it, in the extended CSM, the change of
the sign of current occurs only at t˜ = 0.5, at the point
1045
V Meerovich et al
Figure 8. Calculated temporal dependences of applied and
penetrating magnetic field (a = 1).
Figure 7. Theoretical distribution of current density in
HTSC sample for different times (penetration depth less
than thickness of sample). H̃0 = 1; a = 1.
on the negative slope of the curve of an applied magnetic
field where the electric field reverses its sign. Further,
the penetration process is characterized by two moving
zones.
Note that the penetration depth continues to increase
even after the zone with opposite sign appears. This process
becomes of a diffusional character and lasts as long as the
addition to the critical current attenuates. The attenuation
is due to both the resistive dissipation and the propagation
of the current deep into a superconductor. Finally, two
zones with opposite signs of jc are reached (figure 7(b)).
If the penetration depth is less than the sample thickness,
the areas of the zones in figure 7(b) should be the same. If
the field penetrates through the sample, these areas differ
and the trapped flux appears (figure 2 and figure 8).
Figures 8 and 9 show the calculated results of the
penetration of the magnetic field for our experimental
conditions. As one can see, the calculated curves are in
good agreement with the experimental curves shown in
figures 2 and 3.
1046
Figure 9. Calculated dependences of penetrating magnetic
field on applied magnetic field for the BSCCO 2212
plate-shaped sample (a = 0.5).
5. Discussion
The extended CSM describes well and explains the
experimental dependences.
The breakthrough field
proved to be dependent not only on the superconductor
characteristics jc , 1 and ρf but also on the parameters
of applied magnetic field pulse (amplitude and duration).
We consider only the flux flow region with the linear E–j
relationship. Considering the voltage–current characteristic
of HTSC samples, some authors take into account the
flux creep region with exponential E–j relationship [7].
However, the results of [23, 24] and also our measurements
show that this region ends below 1 µV cm−1 . For our
BSCCO sample, this corresponds to about 0.001 in the
dimensionless units introduced above. This means that we
can neglect the flux creep region and use the linear voltage–
current characteristic in the form of (3).
The peculiarities of penetration of a time-varying
magnetic field into high-Tc superconductors have been
Penetration of magnetic field pulse into HTSC
observed by many authors [4–7]. Nevertheless, numerous
papers on magnetic shielding and penetration under AC
magnetic fields do not give an adequate answer to the
question of a frequency limit of the applicability of the
classical CSM to high-Tc superconductors. For example,
in [20] this model is used to study the magnetic penetration
at a frequency of 1000 Hz. However, our experimental
results presented above show marked deviation from this
model already at 50 Hz.
The theoretical analysis shows that the basic parameter
which determines the degree of deviation from the classical
CSM is the parameter a, that is the square of the ratio of
the plate thickness to the skin depth. When this ratio is
much less than 1, the classical model is applicable. With
increasing frequency or thickness of the superconductor,
the parameter a increases as a square function. Deviations
from the classical CSM become crucial. The discrepancy
existing in the literature between criteria of applicability
of the classical CSM is explained by different values of
the parameter a, specifically by different thicknesses of the
superconducting samples.
Note that the results obtained show that pulse methods
for critical current measurements are of limited usefulness.
The criterion of applicability of pulse methods is a 1.
Acknowledgments
We wish to thank the Ministry of Science and the
Arts of Israel and the Technion Crown Center for
Superconductivity for their support of our work.
References
[1] Bean C P 1962 Phys. Rev. Lett. 8 250; 1964 Rev. Mod.
Phys. 36 31
[2] Dersch H and Blatter G 1988 Phys. Rev. B 38 11 391
[3] Bryksin V V, Goltsev A V and Dorogovtsev S N 1990
Physica C 172 352
[4] Macfarlane J C, Driver R, Roberts R B, Horrigan E C and
Andrikidis C 1988 Physica C 153–155 1423
[5] Fisher L M, Mirkovic J, Voloshin I F, Makarov N M,
Yampol’skii V A, Rodriguez F Perez and Snyder R L
1994 Appl. Supercond. 2 685
[6] Jiang H and Bean C P 1994 Appl. Supercond. 2 689
[7] Uesaka M, Suzuki A, Takeda N, Yoshida Y and Miya K
1995 Cryogenics 35 243
[8] Rhyner J 1993 Physica C 212 292
[9] Bean C P 1989 Superconductivity and Applications ed H S
Kwok et al (New York: Plenum) p 767
[10] Goren S, Jung G, Meerovich V, Sokolovsky V, Skoletsky I
and Homjakov V V 1994 Proc. EUCAS’93, Applied
Superconductivity ed H C Freyhardt (Oberusel:
Deutsche Gesellschaft für Material-Kunde e.V.) p 749
[11] Hole C R J, Jones H and Goringe M J 1994 Meas. Sci.
Technol. 5 1173
[12] Hole C R J, Jones H, Burgoyne J W, Dew-Hughes D,
Grovenor C R M and Goringe M J 1995 IEEE Trans.
Appl. Supercond. 5 1313
[13] Jones H, Hole C R J, Ryan D T and van der Burgt M 1995
Applied Superconductivity (Inst. Phys. Conf. Ser. 148) ed
D Dew-Hughes (Bristol: Institute of Physics) p 89
[14] Ryan D T, Hole C R J, van der Burgt M, Jones H, Davies
C M, Grovenor C R M, Goringe M J and Dew-Hughes
D 1996 IEEE Trans. Magn. 32 2803
[15] Hole C R J, Ryan D, Dew-Hughes D, Jones H, Goringe M
J and Grovenor C R M 1996 Physica B 216 291
[16] Meerovich V, Sinder M and Sokolovsky V 1996
Supercond. Sci. Technol. 9 734
[17] Bock J, Bestgen H, Elschner S and Preisler E 1993 IEEE
Trans. Appl. Supercond. 3 1659
[18] Shter G E and Grader G S 1994 J. Am. Ceram. Soc. 77
1436
[19] Ohshima S and Okuyama K 1990 Japan. J. Appl. Phys. 29
2403
[20] Wang J and Sayer M 1993 Physica C 212 395
[21] Chandran M and Chaddah P 1995 Supercond. Sci. Technol.
8 774
[22] Takeda N, Uesaka M and Miya K 1995 Cryogenics 35 893
[23] Polák M, Koffman P, Majoroš M, Kedrová M and
Plechácek V 1990 Supercond. Sci. Technol. 3 67
[24] Bock J, Elschner S, Herrmann P F and Rudolf B 1995
Applied Superconductivity (Inst. Phys. Conf. Ser. 148) ed
D Dew-Hughes (Bristol: Institute of Physics) p 67
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