PHYSICAL REVIEW B
VOLUME 57, NUMBER 9
1 MARCH 1998-I
Accurate determination of the critical state in anisotropic superconductors
from transport measurements
N. Adamopoulos and S. K. Patapis
Department of Physics, Section of Solid State Physics, University of Athens, Panepistimiopolis GR 157 84, Athens, Greece
~Received 14 October 1997!
A method to accurately determine the critical state and the field profile inside an anisotropic superconductor
is presented, in which the self-field effect is eliminated. A transport current at its critical value is passing along
a slablike superconductor connected in series and placed between two copper slabs of exactly the same cross
section. The perpendicular component of the self-field at the edges of the highly textured superconducting
sample is compensated by the field produced by the copper slabs and a uniform magnetic field results parallel
to the surface of the sample and to the a-b planes of the superconducting grains. This configuration allows the
true critical current along the a-b planes to be measured as a function of the magnetic field aligned purely
parallel to the a-b planes throughout the sample width. Furthermore, features of the critical state can be
deduced from the field dependence of the critical current without using any fitting routines. In particular, when
the critical state is anisotropic, as in the case of a highly textured oxide superconductor with the a-b planes
parallel to the slab surface, this configuration allows the critical current density along the a-b planes to be
measured as a function of magnetic field aligned purely parallel to the a-b planes. @S0163-1829~98!00709-7#
I. INTRODUCTION
As first suggested by Bean, a critical state is established in
an inhomogeneous mixed-state superconductor.1 In the simplest case, the critical current density is independent of the
local magnetic field and is constant inside the superconductor; in a slab geometry the internal field then varies linearly with position. However, the critical state is described
by a specific field dependent critical current density Jc(B),
and the field profile varies with position in a way determined
by Maxwell’s equation ,3B5 m 0 Jc(B), with Jc(B) determined by the pinning mechanism and summation model applicable for the particular material.2 Jc(B) is often deduced
by performing either magnetization measurements or ac field
penetration measurements in superconducting cylinders, and
by using fitting routines to distinguish between different
critical state models, some of them involving three unknown
parameters.3 Here, we present a method that enables the deduction of part of the magnetic-field profile directly from
transport measurements. This method can be applied to any
class of material. However, for anisotropic samples one has
to consider the fact that the critical current density along the
i
a-b planes J ab
and the irreversibility field
c (H ab)
B irr(H i ab) are much higher when the magnetic field is parallel to the a-b planes than when the field is perpendicular to
the a-b planes.4,5 A transport current in a slablike material
with width several times its thickness produces a self-field at
the edges with a strong component perpendicular to the
sample surface.6,7 For a textured material, with a2b planes
parallel to the slab surface, this component is perpendicular
to the superconducting grains. For a thin sample of thickness
2a carrying a current density J, the field is almost perpendicular at the edges and approximately 2Ja. In such a case,
i
the current density is J ab
c (H ab) at the center of the sample
and towards the edges of the sample approaches J ab
c (H'ab)
and therefore the critical current measurement underesti0163-1829/98/57~9!/5055~4!/$15.00
57
mates the true value. Attempts to reduce the self-field effect
have been made in the past by placing two superconductors
side by side, and by studying the field profile for different
degrees of critical current dependence upon the perpendicular field.8 In the method presented here two copper conductors with exactly the same cross section as the superconducting slab are placed on either side of the sample to be
measured, all three conductors carrying the same current and
thus reducing the perpendicular component of the field at the
superconductor. An additional magnetic field applied along
the a-b planes and perpendicular to the current moves the
field profile inside the sample allowing the critical current to
be measured independent of sample size as a function of
field strictly along the a-b planes. It will be demonstrated
that from such a measurement the field profile and the critical state Jc(B) can be studied, with emphasis given to the
value of the critical current density at zero magnetic field.
II. PREPARATION OF SAMPLES
The Bi2Sr2CaCu2Ox samples were prepared by the composite reaction texturing method by mixing superconducting
powder with MgO whiskers and partial melting.9 The whiskers were aligned randomly on the plane of the flat surface of
the sample resulting in a textured microstructure with the
superconducting grains aligned preferably with their c axis
perpendicular to the surface of the sample and the plane of
the whiskers. The texture was confirmed by electron microscopy and pole figure analysis from which it was concluded
that 80% of the grains are aligned with their a-b planes
between 0 and 20° with respect to the sample surface. The
transport critical current density is measured by the fourprobe method with a criterion of 1 m V cm21 at 77 K for a
sample of width 6.48 mm and thickness 0.95 mm with a
magnetic field applied parallel to the sample surface.
5055
© 1998 The American Physical Society
5056
57
BRIEF REPORTS
FIG. 1. Schematic arrangement of the superconductor and the
copper conductors for the reduction of the perpendicular component
of the self-field. All three conductors carry the same current and
have the same dimensions: w56.48 mm, 2a50.95 mm, C
50.2 mm.
III. DESCRIPTION OF METHOD
AND EXPERIMENTAL RESULTS
The superconducting sample and two copper conductors
were arranged as shown in Fig. 1. The magnetic fields of the
individual conductors superpose producing, approximately, a
field which is homogeneous and parallel over the superconductor surface. The self-field measurement therefore involves a reduced z-axis component. The critical current density as a function of magnetic field is shown in Fig. 2 for two
cases: ~1! for all three conductors carrying the same current
by connecting them in series, and ~2! for the superconducting
sample connected to the current supply. In the first case the
critical current is higher than the second by more than 20%,
at zero applied field. At high magnetic fields the two curves
are almost indistinguishable and decrease slowly with field.
Similar characteristics have been observed in single crystals
and have been attributed to the field penetration above the
lower critical field H c1 . 10 In the next paragraph we will
demonstrate that this behavior is a direct result of the form of
the field profile and the critical state inside the superconductor.
FIG. 2. Critical current as a function of applied magnetic field
for a polycrystalline sample of width 6.48 mm and thickness 0.95
mm. The magnetic field was applied parallel to the sample surface
and perpendicular to the current and the superconductor was placed
between two copper slabs of exactly the same cross section. Graph
~1! is for all three conductors carrying the same current and in the
same direction and graph ~2! for only the superconducting sample
connected to the power supply.
FIG. 3. An hypothetical field profile and its evolution under an
applied magnetic field parallel to the surface of the sample and
perpendicular to the current. The field profile is shown ~a! for the
self-field situation, ~b! for a small increment of the magnetic field
from zero, and ~c! at a characteristic value H * of the applied field,
where the field at one surface of the sample becomes zero and ~d! at
larger fields.
IV. DISCUSSION
At zero applied magnetic field, the self-field of a superconductor when the transport current is at its critical value
has a complex distribution, bending around the edges, depending on the shape of the conductor’s cross section. For an
infinitely thin conductor at y50 the self-field is directed parallel to the sample surface and its value can be derived using
Ampere’s law: 2Hw5I, w the width of the sample. The
same simple relation can also be used for the case of a conductor with a circular cross section. For any other geometry
however the situation is more complex and for a slab geometry the self-field peaks at the edges of the sample, where it
turns around the cross section. In the case of an anisotropic
high T c material textured with the a-b planes parallel to the
slab surface, the z component of the self-field reduces the
local value of the critical current density. This self-field effect has been studied in the past by measuring the value of
the critical current as a function of the sample thickness as
the material is progressively thinned.11 Although this method
demonstrates that the self-field does affect the transport critical current, it is not quantitative, since samples with different
thickness correspond to a different z-axis component and
spatial variation of the self-field.12 In the present method the
self-field is uniform across the width both in direction and
magnitude and the field profile inside the material will be
i
determined by the field dependence of J ab
c (H ab) only. A
schematic profile is shown in Fig. 3 that qualitatively shows
the main properties of the critical state: the current density
dH y /dz is maximum at the center (H y 50) and minimum at
the surface @ H y 5H s f (a) # . Curve (a) corresponds to the
self-field situation, while curves (b), (c), and (d) correspond to different values of the applied field directed also
parallel to the sample surface. Curve (b) is for a small increment of the magnetic field from zero, in which case the
field profile is uniformly displaced along the z axis. At some
characteristic value H * of the applied field, the field at one
surface of the sample becomes zero, curve (c), and the field
profile at that surface is very sensitive to a small change of
the magnetic field, since its slope is very sharp. At larger
fields the field profile has the shape shown by curve (d) in
Fig. 3.
Assuming that the magnetic field at one surface is H A
57
BRIEF REPORTS
5057
FIG. 5. A hypothetical field profile, f (z), inside a superconducting sample at zero applied parallel magnetic field. An external magnetic field moves the field profile by an amount d; the field profile
can then be described by the mathematical function f (z1 d ). At
some characteristic value of external field, the field profile is described by f (z1a) and the magnetic field at one edge becomes
zero.
FIG. 4. ~a! A Jc(B) plot for a critical state model described by
J c 5J ` /(11H/H o ) n for J ` 53 107 Am22, H o 5103 Am21, and n
51. ~b! DH vs H appl dependence for such a Jc(B) model.
5H(z52a) and H B 5H(z5a) at the other when the applied
magnetic field is H appl , the measured critical current will
simply be
I c 5H B w2H A w5w ~ H B 2H A ! 5wDH, or I c /w5DH
~1!
with DH5H B 2H A . Correspondingly, the applied field can
be written as
H appl5 ~ H B 1H A ! /2.
~2!
For a Bean critical state DH and therefore I c will remain
the same, however for any other critical state DH and I c will
progressively decrease. When H appl reaches H * at some
critical point, the magnetic field H A will become zero, with
H appl5DH/2. At that point, the variation of DH ~or equivalently I c ! with H appl will show a point of inflection, since, for
any critical state other than Bean’s, any change of the applied field will significantly affect H A while leaving H B
much less affected. At higher field inductions, the magnetic
fields H A and H B will approach each other, resulting in a
gradual reduction of the critical current falling slowly with
field.
One can predict the form of the DH vs H appl for any given
critical state Jc„B…. Figure 4~a! shows the field dependence
of the critical current density described by J c 5J co /(1
1H/H o ) n for J co 53 107 Am22, H o 5103 Am21, and n
51. The field profile for any value of the applied field can be
easily found, leading to a simple evaluation of H A and H B .
In Fig. 4~b!, the DH vs H appl dependence for the above criti-
cal state model is shown. This demonstrates that a fitting
routine can always be used to deduce the form of Jc(B)
which predicts the experimental DH vs H appl results.
A further insight into the problem can be given by noticing that the field distribution inside the sample at self-field
conditions possesses some characteristic properties. This
field distribution, described by the function f (z) in Fig. 5,
once determined, can describe the properties of the magnetic flux distribution, at all levels of the applied field. If
J A and J B are the current densities at the two surfaces
~J A 5d f (z1d!/dz u z52a and J B 5d f (z1 d )/dz u z5a ), from
Eqs. ~1! and ~2! we have
d ~ DH !
J A 2J B
52
,
dH appl
J A 1J B
~3!
which can also be written as
JB
4
215R ~ H appl! .
5
J A d ~ DH !
12
dH appl
~4!
Equation ~3! shows that the quantity d(DH)/dH appl can attain a maximum absolute value of 2 at some field H appl
5H * at which H A 50, for the extreme case of a critical
current density J A at point A much larger than J B . Equation
4 also expresses that R(H appl), which is a measured quantity
and gives the ratio between J A and J B at any instant. At the
point of inflection, where H appl5H * and DH52H * , the
magnetic field H B at point B will be 2H * and that at point
A, H A will be 0, and therefore Eq. ~4! becomes
J ~ 2H * ! 5R ~ H * ! J ~ 0 ! .
~5!
Through a series of expansions applied to arbitrary field profile one can find an expression which holds for any realistic
critical state model for the ratio of two defined current densities at A and B, J(0) and J(DH max/2):
F
J ~ 0 ! 5J ~ DH max/2! 12
1 d ~ DH !
2 dH appl
U
H appl5H *
G
.
~6!
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BRIEF REPORTS
Equations ~5! and ~6! give the field dependence of the
critical current density at different field values normalized
with respect to its value at zero magnetic field, once the field
dependence of the critical current is known for an applied
magnetic field aligned strictly parallel to the a-b planes. In
applying the above analysis to a real material, we have to
bear in mind that no sample is perfectly textured, and there
are always going to be inhomogeneities like porosity, cracks,
etc. However, for an ideal sample, the above treatment is, to
our knowledge, the only way to deduce accurately the magnetic field dependence of J c directly from transport measurements. Other methods include magnetization measurements
from which the critical current density is calculated given the
width of the hysteresis loop; such a method assumes that the
current length scale is known, otherwise intergranular and
intragranular currents contribute altogether to the magnetic
moment of the sample. Another method consists of measuring the flux through a superconducting cylinder and fitting
the results with a general formula in which three unknown
parameters are allowed to vary to achieve the best fit with the
experimental data. The same procedure can be used in the
transport measurements presented here, however the above
analysis ha shown that properties and features of the field
profile can be deduced without using any fitting routines.
The above analysis helps in correcting the horizontal axis
of Fig. 2, with the magnetic field being a result of both the
applied field and the self-field and directed parallel to the
a-b planes. Further analysis according to Eq. ~4! can provide
more information on the full J c (H) characteristic of the material. This analysis will be presented in a future publication.
The above treatment has implications for a wide class of
materials and measurement configurations. The self-field effect is expected to be enhanced in samples with thickness
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1
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57
comparable to their width, in which the field nonuniformity
extends deep inside the material and away from the edges. In
thin films, the edges are subjected to a stronger field than the
middle part, and the strong perpendicular component will
severely affect the depinning of vortices.13 These issues have
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from different samples and under different conditions.
V. CONCLUSIONS
In conclusion, we have measured the critical current density of an anisotropic polycrystalline sample under a magnetic field aligned parallel to the a-b planes of the grains, by
cancelling the perpendicular component of the self-field.
This is particularly important at low fields, where the transport current is high and the produced self-field is comparable
to the applied field, and the z-axis component is very strong
at the edges of the sample. We have also demonstrated that it
is possible to deduce properties of the critical state J c (H) by
analyzing the I c vs H appl experimental data. A further, more
detailed, analysis will be presented in an extended future
publication. The method given here can be a simple and
valuable tool for characterizing isotropic and anisotropic
samples independent of their dimensions. It can also be used
in characterising thin-film samples where applying a magnetic field at different angles with respect to transport current
changes the field orientation and boundary conditions at the
edges.
ACKNOWLEDGMENT
The authors would like to thank Dr. J. E. Evetts for valuable discussions on the present work.
9
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