Physica C 315 Ž1999. 241–246
Distribution of flux pinning strength in a superconducting
Bi-2223 silver-sheathed tape
M. Kiuchi
a
a,)
, A. Yamasaki b, T. Matsushita
a,b
, J. Fujikami c , K. Ohmatsu
c
Graduate School of Information Science and Electrical Engineering (ISEE), Kyushu UniÕersity, 6-10-1 Hakozaki, Higashi-ku,
Fukuoka 812-8581, Japan
b
Department of Computer Science and Electronics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502, Japan
c
Basic High Technology Laboratories, Sumitomo Electric Industries, 1-1-3 Shimaya, Konohana-ku, Osaka 554-8511, Japan
Received 19 February 1999; accepted 18 March 1999
Abstract
The distribution of the flux pinning strength was estimated from a measured current–voltage curve for a monofilamentary
Bi-2223 silver-sheathed tape in a magnetic field parallel to the c-axis. It is found that the distribution can be fitted well to
the Weibull function of a very wide distribution. This wide distribution seems to be caused by the low critical current density
at weak links. The obtained Weibull function can be approximated fairly well by a simpler distribution function with which
the scaling of current–voltage curves is analyzed using the flux creep–flow model. The obtained scaling parameters agree
well with the experimental results. q 1999 Elsevier Science B.V. All rights reserved.
Keywords: Flux pinning strength; Bi-2223 silver-sheathed tape; Weibull function
1. Introduction
It is important to clarify the electromagnetic properties in high-temperature superconductors for their
application to power equipments. These properties
are strongly influenced by the dynamic behavior of
fluxoids in the superconductor and can be investigated in detail by measuring the current–voltage
curves. It is empirically known that the current–voltage curves can be scaled by normalizing with suitable functions of temperature. Such a behavior is
explained by the two mechanisms. One of them is
the vortex glass–liquid transition w1–3x and the other
)
Corresponding author. Tel.: q81-092-642-3893; Fax: q81092-642-3963; E-mail: kiuchi@ees.kyushu-u.ac.jp
is the flux creep and flow w4,5x. However, it was
recently shown w6x that the thermal depinning of
fluxoids resulted from the flux creep is the secondorder transition and that it is identical with the vortex
glass–liquid transition. That is, the fluxoids can be
regarded to be in the so-called glass state when the
flux pinning is effective even under the influence of
flux creep. This theoretical result is consistent with
various experimental results. The details of comparison with experiments are given in Ref. w6x. One of
the important points is that the distribution of flux
pinning strength influences remarkably the current–
voltage curves. For example, it is necessary to assume a very wide distribution to explain the characteristic current–voltage curves in high-temperature
superconductors with the low n-value, which is de-
0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 2 4 6 - 4
242
M. Kiuchi et al.r Physica C 315 (1999) 241–246
fined by expressing the relationship between the
electric field and the current density as E A J n w5,7x.
This seems to be caused by weak links at grain
boundaries, which bring about very low local critical
current density. On the other hand, such a current–
voltage curve in the flux flow regime is caused by
the distributed local critical current density and it is
known that this distribution can be described by the
Weibull function w8,9x, which is frequently used in
the field of reliability engineering.
In this study, the Weibull function is estimated
from the current–voltage curve for a Bi-2223 silversheathed tape under a magnetic field parallel to the
c-axis. A simpler distribution function used in the
flux creep–flow model is compared with the obtained Weibull function. In addition, the scaling of
the current–voltage curves estimated from the flux
creep–flow model using the simpler distribution
function is compared with the experimental results.
2. Experiments
The specimen used in this study was a silversheathed Bi-2223 monofilamentary tape prepared by
the powder-in-tube method. The critical temperature
estimated using a SQUID magnetometer was 108.1
K. The typical cross-section of the specimen was
approximately 3.4 mm wide and 0.17 mm thick, and
Fig. 1. Scaling of observed current–voltage curves at B s 0.5 T.
Fig. 2. Experimental Žv . and theoretical results Ž`. of transition
line.
the thickness of the superconducting region was
approximately 50 mm. The c-axis was approximately directed normal to the flat surface of the tape.
The current–voltage curves were measured under
a magnetic field parallel to the c-axis using the four
terminals method. The length of the measured specimen was approximately 45 mm and the voltage was
measured across the potential leads separated by 10
mm. A pulsed transport current with a width of 1 s
Fig. 3. Experimental Žv . and theoretical results Ž`. of static
critical index, Õ.
M. Kiuchi et al.r Physica C 315 (1999) 241–246
Fig. 4. Experimental Žv . and theoretical results Ž`. of dynamic
critical index, z.
was applied to the specimen to decrease the Joule
heat at current leads. The temperature of the specimen was controlled in the range of 10–84 K in an
atmosphere of helium gas. The stability of the temperature during the measurement was about "0.1 K
at low current density and about "0.5 K at high
current density. The details of the measurement are
described in Ref. w7x.
The scaling of the measured current–voltage
curves was examined using the scheme of vortex
glass–liquid transition theory. The scaled result at
B s 0.5 T is shown in Fig. 1. The obtained transition
temperature, static and dynamic critical indices are
Tg s 50 K, Õ s 0.85 and z s 8.8, respectively. The
magnetic field dependence of these scaling parameters is shown in Figs. 2–4. It is found that z increases and Õ decreases with increasing magnetic
field. Such tendencies of the critical indices are
similar to the case of Bi-2212 w10,11x and are ascribed to the inhomogeneous distribution of the critical temperature w7x.
3. Discussion
In high-temperature superconductors, it is known
that the voltage does not drop off sharply with
decreasing current as characterized by the low n
243
value even at low temperatures andror at low magnetic fields. This result suggests that the flux pinning
strength in high-temperature superconductor is distributed over a wide range. In fact, the experimental
results of the scaling of current–voltage curve for
Bi-2223 tapes was explained well by the flux
creep–flow model in which the flux pinning strength
was assumed to be distributed over one order of
magnitude and a half w7x.
It is known that the distribution of the pinning
strength can be approximately expressed in the form
of the Weibull function. In this case, the distribution
can be estimated from the measured current–voltage
curve using the Weibull analysis w8,9x with the assumption that the observed voltage comes only from
the flux flow. Here, the distribution of pinning
strength in the present tape specimen is investigated
using this method. We assume that at relatively low
fields, the scaling law for the virtual critical current
density, Jc0 , in the creep-free case is given in the
empirically known form:
Jc 0 s A 1 y
T
2
m
ž /
Tc
B gy1 ,
Ž 1.
where A, m and g are pinning parameters. In the
above equation, A is equal to the critical current
density at 0 K and 1 T, and represents the flux
pinning strength of the superconductor. Since the
distribution of the pinning strength is very wide in
high-temperature superconductors, we assume for
simplicity that only A is distributed in the form of
the Weibull function:
X
f Ž A. s
mX
D
X
X
ž
A y A0
D
m y1
/
exp y
ž
A y A0
D
m
/
,
Ž 2.
where A 0 is the minimum value of A, D represents
the width of the distribution and mX is a parameter
determining the shape of the distribution.
The measured current–voltage curve at T s 54 K
and B s 0.2 T is shown by solid symbols in Fig. 5.
The curvature of this current–voltage curve is convex, and the effect of flux creep is not significant in
244
M. Kiuchi et al.r Physica C 315 (1999) 241–246
On the other hand, the distribution function of the
flux pinning strength is assumed to be simply expressed as:
f Ž A . s K exp y
Fig. 5. Current–voltage curve at T s 54 K and B s 0.2 T. Symbols and line represent experimental result and theoretical expression using the Weibull function, respectively.
the obtained current–voltage curve. Therefore, we
assume here that the measured distribution of flux
pinning strength is the same as that of A and that the
measured critical current density obeys the same
scaling law as Jc0 in Eq. Ž1.. This will be allowed
since the magnetic field is very low, although the
temperature is relatively high. The Weibull parameters can be determined by fitting the theoretical
curve with the observed result and we have A 0 s 5.2
= 10 7, D s 7.0 = 10 8 and mX s 1.7, where the used
pinning parameters are m s 3.9 and g s 0.70 as will
be argued later. The theoretical current–voltage curve
is compared with the observed result in Fig. 5. These
pinning parameters are determined so that a good fit
is obtained between the observed critical current
density and the theoretical result of the flux creep–
flow model. A slight deviation at low electric fields
is considered to result from the flux creep.
The distribution of the flux pinning strength expressed using the Weibull function is shown by the
chained line in Fig. 6. It is seen that the distribution
of the flux pinning strength in the Bi-2223 silversheathed tape is very wide and its width extends over
an order of magnitude.
Ž log A y log A m .
2s 2
2
Ž 3.
in the flow creep–flow model w5x. In the above, A m
is the most probable value, s 2 is a constant representing the degree of deviation and K is a constant
determined by the condition of normalization. The
advantage of the assumption of Eq. Ž3. is that the
number of adjusting parameters is smaller than that
of Eq. Ž2.. If A m and s 2 are determined so as to get
a good fit between Eqs. Ž2. and Ž3., we have s 2 s
0.092 and A m s 3.93 = 10 8. The distribution function with these parameters is shown by the solid line
in Fig. 6. It is seen that the two distribution functions
agree very well with each other. The values of s 2
obtained similarly at 46 and 60 K are 0.093 and
0.095, respectively. Thus, the value of s 2 is almost
independent of temperature.
Here, we shall theoretically reanalyze the scaling
of E–J curves using the above parameters. The
details of the analysis of the current–voltage curve
using the flux creep–flow model are described in
Ref. w5x. The superconducting and pinning parame-
Fig. 6. Comparison of two distribution functions of critical current
density. Solid and chained lines represent Eqs. Ž2. and Ž3.,
respectively.
M. Kiuchi et al.r Physica C 315 (1999) 241–246
Fig. 7. Scaling of calculated current–voltage curves at B s 0.5 T.
ters used in the numerical calculation are; Tc s 108.1
K, Bc2 Ž0. s 50.0 T, rnŽTc . s 100 mV m, A m s 3.98
= 10 9 , s 2 s 0.092, m s 3.9 and g s 0.70. The
number of flux lines in the flux bundle, g 2 , related
to the transverse flux bundle size, is an important
parameter for estimation of the pinning potential. It
is considered that g 2 is determined so that the
critical current density under the flux creep might be
maximized w12x. g 2 is estimated as 0.97 at T s 60 K
and B s 0.60 T. Similar results are obtained in the
whole region of the present analysis. However, since
the minimum value of g 2 is 1, we use g 2 s 1. The
numerical analysis was carried out using the above
parameters.
Fig. 7 shows the scaled results of numerically
calculated curves. Comparing this with Fig. 1, it is
seen that the agreement is quite satisfactory. The
scaling parameters obtained from the numerical analysis are Tg s 52 K, Õ s 0.89 and z s 8.4, and these
are close to the experimental values ŽTg s 50 K,
Õ s 0.85 and z s 8.8.. In Figs. 2–4, the experimental and theoretical results are compared for the transition line, Õ and z, respectively. The agreements are
also good for these results. Hence, the assumed wide
distribution of the flux pinning strength over an
order of magnitude seems to be reasonable. Since
similar distribution width is needed to explain the
scaling behavior in Bi-2212 tapes w10,11x, such a
wide distribution seems to be inherent to Bi-based
245
superconductors. The above argument suggests that
the present simpler function can express the characteristic distribution of the local critical current density in these materials.
Recently, it was reported that the distribution
function of the flux pinning strength estimated from
the second derivative of the voltage vs. current curve
had two peaks in a dip-coated Bi-2212 superconducting tape w13x. The peak at poor pinning strength
seems to be caused by weak links at inhomogeneous
regions. The resultant scaling was appreciably different from others with single peak distributions: the
obtained z was very small, suggesting a low n
value. That is, the current–voltage curves in hightemperature superconductors are remarkably influenced by the inhomogeneous distribution of the flux
pinning strength. In the above, we have assumed for
simplicity that only A in Eq. Ž1. is distributed but Tc ,
m and g are not. The recently observed result that z
increases and Õ decreases with increasing magnetic
field seems to be caused by the inhomogeneous
distribution of the critical temperature w7x. Thus, it is
necessary to investigate the influence of these inhomogeneous distributions in more detail to understand
the current–voltage curves.
4. Summary
The distribution of the flux pinning strength under
a magnetic field parallel to the c-axis was analyzed
using the Weibull function for a monofilamentary
Bi-2223 silver-sheathed tape. The obtained distribution function could be approximated by a simpler
distribution function, with which the current–voltage
curve was calculated using the flux creep–flow
model. Theoretical results on the transition line and
the two critical indices agreed well with the experimental results. This agreement suggests that the distribution of the flux pinning strength over an order of
magnitude is reasonable and that the simpler distribution function is useful for description of the flux
dynamics.
References
w1x M.P.A. Fisher, Phys. Rev. Lett. 62 Ž1989. 1415.
w2x D.S. Fisher, M.P.A. Fisher, D.A. Huse, Phys. Rev. B 43
Ž1991. 130.
246
M. Kiuchi et al.r Physica C 315 (1999) 241–246
w3x R.H. Koch, V. Foglietti, W.J. Gallagher, G. Koren, A. Gupta,
M.P.A. Fisher, Phys. Lett. 63 Ž1989. 1511.
w4x T. Matsushita, N. Ihara, Proc. 7th Int. Workshop on Critical
Currents in Superconductors, World Scientific, Singapore,
1994, p. 169.
w5x T. Matsushita, T. Tohdoh, N. Ihara, Physica C 259 Ž1996.
321.
w6x T. Matsushita, T. Kiss, Physica C 315 Ž1999. 12.
w7x M. Kiuchi, K. Noguchi, T. Matsushita, T. Kato, T. Hikata,
K. Sato, Physica C 278 Ž1997. 62.
w8x F. Irie, Y. Tsujioka, T. Chiba, Supercond. Sci. Technol. 5
Ž1992. S379.
w9x T. Kiss, T. Nakamura, H. Takeo, K. Kuroda, Y. Matsumoto,
F. Irie, IEEE Trans. Appl. Supercond. 5 Ž1995. 1363.
w10x M. Kiuchi, K. Noguchi, M. Tagomori, T. Matsushita, T.
Hasegawa, IEEE Trans. Appl. Supercond. 6 Ž1997. 2114.
w11x K. Noguchi, M. Kiuchi, M. Tagomori, T. Matsushita, T.
Hasegawa, Advances in Superconductivity IX, Springer,
Tokyo, 1997, p. 625.
w12x T. Matsushita, Physica C 217 Ž1993. 461.
w13x Y. Nakayama, Y. Takase, M. Kiuchi, T. Matsushita, T.
Hasegawa, Physica C 315 Ž1999. 235.