# Introduction of Matsushita Labo.

Keywords: Superconducting, Flux Pinning, Critical Current Density, High-Tc superconductor
Main International Conference:

Introduction text from ISTEC Journal Vol. 8 No. 1 1995


0. Introduction

Four graduate students in the master course and four under
graduate students are studying the flux pinning and related
electromagnetic phenomena in superconductors with Prof.
Matsushita and Mr. Otabe.  They contribute also to the study
of magnetic bearing system using high temperature
superconductors which is conducted by Dr. Mochimutsu Komori
in Department of Mechanical Systems Engineering in the same
faculty.  The main topics in this group are (1) the
mechanism of flux pinning and the summation problem, (2)
evaluation of critical current characteristics in
superconductors, (3) electromagnetic phenomena related to
the fluxoid motion and (4) the longitudinal field effect.
In the following the main points of these topics are
introduced.

1. The mechanism of flux pinning and the summation problem

(a)elastic moduli of fluxoid lattice

The macroscopic pinning force density depends not only on
the strength and the concentration of pinning centers but
also on the elastic moduli of pinned fluxoids.  There are
three independent moduli, i.e., the uniaxial compression
modulus $C_{11}$, the tilt modulus $C_{44}$ and the shear
modulus $C_{66}$.  As for $C_{11}$ and $C_{44}$, two
theories were proposed.  One is the local theory of
Labusch$^{4)}$ in which these moduli are predicted to be
independent of the wave number of deformation of the fluxoid
lattice and the other is the nonlocal theory of
Brandt$^{5)}$ in which these moduli are predicted to be
dependent.  It is considered that the result of the nonlocal
theory is doubtful, since the Maxwell stress tensor which
derives the Lorentz force is independent of the wave number,
while the Lorentz force is also expressed in terms of these
elastic moduli.  Then, it was shown$^{6)}$ that the
quantization of magnetic flux and ${\rm div}\mathb{B}=0$ are
not satisfied in the derivation of $C_{11}$ and $C_{44}$,
respectively, in the nonlocal theory.  At the same time, the
local result was found to satisfy these requirements.

(b)flux pinning characteristics in Nb-Ti

The highest performance in the flux pinning efficiency is
achieved in Nb-Ti among commercial superconductors.  The
dominant pinning centers in Nb-Ti is normal $\alpha$-Ti
phase of the shape of thin ribbons.  In the case of thin
normal layers comparable to or smaller than the coherence
length, the superconductivity appears even in the normal
region due to the proximity effect.  Then, Kramer and
Freyhardt$^{7)}$ insisted that such thin layers are not
effective as pinning centers.  However, this prediction
contradicts with the strong pinning in Nb-Ti and it was
clarified$^{8)}$ in terms of G-L equations that the
elementary pinning force is not reduced appreciably even
under the remarkable proximity effect.  On the basis of this
theoretical result, the pinning characteristics in
commercial Nb-Ti was analyzed$^{9)}$ and the pinning force
in Nb-Ti with artificially introduced pinning centers is
estimated in collaboration with Furukawa Electric Co.
Ltd.$^{10)}$

(c)the summation problem in the regime of strong pinning

It is empirically known that the linear summation, i.e., the
proportionality of the pinning force density to the product
of the pin concentration and the elementary pinning force,
holds for the case of strong pinning such as the pinning by
normal precipitates or grain boundaries.  As for the
theoretical prediction for such a strong pinning property,
the theory of Larkin and Ovchinnikov$^{11)}$ was proposed.
However, this result does not agree with the experiments.
Then, a new statisitcal theory is established in which the
Labusch parameter is used as a coherent parameter
representing the strength of interference of individual
pinning forces and the fact that the long range order does
not exist in the fluxoid lattice as pointed out in the
Larkin-Ovchinnikov theory.  According to the result of this
theory, while the threshold value for the elementary pinning
force exists formally, its value is always smaller than the
elementary pinning force.  Therefore, it is possible to
explain simultaneously$^{12)}$ that the practical threshold
problem does not extist and that the pinning loss is of the
hysteresis type as known well.  The resultant pinning force
density satisfies the linear summation and agrees
quantitatively with experimental result.

2. Evaluation of critical current characteristics in
superconductors

This group has carried out the estimation of critical
current density and the evaluation of weak link properties
in high temperature superconductors$^{13)}$ by measuring the
dependences of critical curernt density on temperature,
history of application of magnetic field and strength of
longitudinal magnetic field.  For this purpose the AC
inductive measurements (Campbell's method) and the AC
susceptibility measurements have been used.  Recently it was
clarified that the reversible fluxoid motion inside the
pinning potential becomes sometimes dominant in high
temperature superconductors, since the small effective size
of superconductor caused by cracks or weak links restricts
severely the region of fluxoid motion.  It was
warned$^{14)}$ that the critical current density is likely
to be overestimated largely if such results of AC
measurements are analysed using the irreversible critical
state model.  The details of this investigation are given in
references.

3. Electromagnetic phenomena related to the fluxoid motion

The electromagnetic phenomena in superconductors are
determined by the fluxoid motion.  One of them is the loss
energy.  In most cases the loss energy in superconductors
can be calculated in terms of the critical state model.
This is correct only when the phenomenon is completely
irreversible.  However, when the superconducting filaments
in multifilamentary metallic superconducting wires becomes
very thin, the loss energy takes much smaller values than
the theoretical estimate because of the reversible fluxoid
motion mentioned in 3.2.  In this group the loss energy
under a remarkable reversible motion of fluxoids is
theoretically estimated$^{15)}$.

The flux creep is also the phenomenon determined by the
motion of fluxoids.  This brings about the relaxation of the
magnetization or the superconducting persistent current,
degrades the critical current density and sometimes reduces
it to zero.  This group has carried out the theoretical
analysis from the viewpoint of flux creep on the
irreversibility line on which the critical current density
is reduced to zero.  Figure 2 represents the irreversibility
lines in various high temperature superconductors and (a) is
experimental results and (b) is theoretical results$^{16)}$.
It was shown that the flux pinning strength and the
two-dimensionality of superconducting materials play the
important roles in determination of the irreversibility
line.  The scaling behavior of the pinning force density in
high temperature superconductors can also be analyzed in
terms of the flux creep model$^{17)}$.  It is also possible
to foresee the characteristics in a material with
artificially introduced pinning centers which will be
fabricated in the future by using this model.

4. Longitudinal field effect

Under the so-called longitudinal magnetic field where the
magnetic field and the current are parallel to each other,
the critical current density takes much larger values than
in the ususal transverse magnetic field, Josephson's
relation $\mathb{E}=\mathb{B}\times\mathb{v}$ does not hold
($\mathb{E}$, $\mathb{B}$ and $\mathb{v}$ are the electric
field, the magnetic flux density and the fluxoid velocity,
respectively) and a negative potential drop is sometimes
observed in the dirction of the current.  It was found that,
when the force-free current parallel to the fluxoids flows,
the fluxoids have a rotationally shearing distorsion as
shown in Fig. 3 and the restoring moment works on the
fluxoids to reduce this distorsion$^{18)}$.  Various
characteristic features under the longitudinal field
geometry can be attributed to the fluxoid motion driven by
this restoring moment.  That is, the rotational fluxoid
motion obtained from the continuity equation of fluxoids is
regarded as a result of the restoring moment and leads to
the electric field deviating from Josephson's
relation$^{19)}$.  The theoretical result of Josephson that
the force-free state is an equilibrium state contradicts the
above interpretation.  This contradiction was found$^{20)}$
to be caused by the choice of incorrect gauge in the theory
of Josephson.  The negative potential drop in the resistive
state was also successfully explained by the model of
fluxoid motion caused by the moment$^{21)}$.  Now the group
is trying to prove the existence of the theoretically
predicted moment which is independent of the Lorentz force
in terms of a torque meter.

References

1)  T. Yasuda, S. Takano and L. Rinderer: Physica B 194-196 (1994) 2235.
2)  T. Yasuda, S. Takano and L. Rinderer: Physica C 208 (1993) 385.
3)  R. Kleiner and P. M ller: Phys. Rev. B 49 (1994) 1327.
4)  R. Labusch: Phys. Status Solidi 19 (1967) 715.
5)  E. H. Brandt: J. Low Temp. Phys. 26 (1977) 709, 735.
6)  T. Matsushita: Physica C 220 (1994) 172.
7)  E. J. Kramer and H. C. Freyhardt: J. Appl. Phys. 51 (1980) 4903.
8)  T. Matsushita: J. Appl. Phys. 54 (1983) 281.
9)  E. S. Otabe and T. Matsushita: Cryogenics 33 (1993) 531.
10)  K. Matsumoto et al.: Appl. Phys. Lett. 64 (1994) 115.
11)  A. I. Larkin and Yu.  N. Ovchinnikov:  J. Low Temp.  Phys.  34 (1979) 409.
12)  T. Matsushita: submitted to 1994 Int. Symp. Supercond., Kitakyushu.
13)  T. Matsushita et al.: IEEE Trans. Appl. Supercond. 3 (1993) 1045.
14)  T. Matsushita, E. S. Otabe and B. Ni: Physica C 182 (1991) 95.
15)  T. Matsushita et al.: Adv. Cryog. Eng. Mater. (Plenum, 1994) p. 551.
16)  T. Matsushita and N. Ihara:  Proc.  Europ.  Conf.  Appl.  Supercond. (1993) p. 779.
17)  T. Matsushita et al.:  submitted to 1994 Int.  Symp.  Supercond., Kitakyushu.
18)  T. Matsushita: J. Phys. Soc. Jpn. 54 (1985) 1054.
19)  T. Matsushita, Y. Hasegawa and J. Miyake:  J. Appl.  Phys.  54 (1983) 5277.
20)  T. Matsushita: Phys. Lett. 86A (1981) 123.
21)  T. Matsushita and F. Irie: J. Phys. Soc. Jpn. 54 (1985) 1066.



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