CHAPTER VI
FLUX CREEP-FLOW MODEL
Edmund Soji Otabe and Masaru Kiuchi

Faculty of Computer Science and Systems Engineering,
Kyushu Institute of Technology,680{4 Kawazu, Iizuka 820{8502, Japanotabe@cse.kyutech.ac.jp

I.

INTRODUCTION

Since oxide superconductor is used at high temperatures and ux pinning isweak, thermal agitation of ux lines is signi cant resulting in high relaxation ofshielding current compared with conventional metal superconductor. For example,
when magnetic eld is trapped by bulk oxide superconductor, the trapped magnetic
eld decreases according to time. This e ect is known as ux creep and rstly explained by Anderson and Kim[1]. That is, the trapped ux line by pinning center isthermally agitated and it could move to the direction of Lorentz force. This causes
reduction of the trapped magnetic eld and the shielding current. When current
density in superconductor is suciently small, ux lines move time to time ( uxcreep) and very small electric eld is induced. Therefore it is not continuous ow
but discontinuous hopping. On the contrary, when the current density becomes high,continuous movement of the ux lines ( ux ow) is dominant.
Electromagnetic properties such as, characteristics of electric eld and current
density (E -J ), critical current density (Jc), irreversibility eld (Bi) in oxide superconductor are well explained by ux creep- ow model. In this chapter, the ux
creep phenomenon is rstly explained and then, numerical and analytical calculations based on ux creep- ow model is described. Finally, some experimental results
are compared with prediction by the ux creep- ow model.
II.

FLUX CREEP PHENOMENON

II.1. Flux creep
When magnetic eld is applied to a superconductor, magnetic eld penetrates
to the superconductor and ux lines are pinned by pinning center. At the same time,ux line tends to move by Lorentz force. These condition is schematically shown in
Fig. 1. The energy is periodically changed and decreasing according to increasing theposition, x. The gradual decrease is representing the work done by the Lorentz force.
In this gure, ux lines move in group unit and this group is called a ux bundle.
According to ux creep model, the possibility to jump over an energy barrier(activation energy), U , for one trial of ux bundle is given by exp(0U=KBT ) which is
known as Arrhenius expression. Here, kB is Boltzmann constant and kBT is thermal

U0
U’
U

af
x

Fig. 1. washboard pinning potential.
energy. If ux bundle succeeded to jump over the activation energy, hopping distanceis considered to be given by ux line spacing, af . According to the movement of theux bundle, electric eld is induced as
" 
!#
U 
U0
E = Baf 0 exp 00 exp 0 k T ;
(1)
kB T
B
where 0 is attempt frequency of the ux bundle, and U 0 is activation energy of
opposite direction to Lorentz force.
Here, we shall calculate a simple case of the relaxation of magnetization by theux creep. A large slab superconductor (0  x  2d) is placed in external magnetic
eld, He along z -axis. From the symmetry, we consider only the half region of 0 x  d. In the case of increasing magnetic eld, induced current ows along y -axis andmovement of the ux bundle by the ux creep is along x-axis. If the current density,
J is assumed to be uniform, magnetic ux density is given by B = 0(He 0 Jx).
Therefore, observed electric eld, E , at the surface of superconductor (x = 0) is
calculated from Maxwell equation, rotE = 0@ B=@t and is given by@dhB i0 d2 @J
E=
=
0
(2)
@t
2 1 @t ;where hB i is the average of magnetic ux density. If we have the relations, U (J )and U 0(J ) as function of J , relaxation of J as a function of time is obtained bysubstituting Eq. (1) to Eq. (2).
When the relaxation is small, the e ect of the ux creep is small. Then it is
supposed U  U 0 and the second term of Eq. (1) can be neglected. Since U decreases
with increasing of J , U can be expanded to
U = U03 0 rJ:
(3)

U

U0

U0*

0

0

1

j

Fig. 2. Relation between activation energy, U , and normalized current density,
j = J=Jc0.
Here U03 is the energy at J ! 0 and is called the apparent pinning potential as shownin Fig. 2. U03, which is experimentally measured in relaxation rate of magnetization,is di erent from true pinning potential U0. Because U0 is only observed at J ! 0
and Eq. (3) is useful at J  Jc0, where Jc0 is virtual critical current density obtainedat U = 0. In other words, virtual critical current density is observed at the case ofux creep free. Therefore, it is approximately considered r = U03=Jc0, and U is given
by

J 
:
(4)
U = U03 1 0
Jc0
Hence Eqs. (1), (2) and (4), we have@J
2
Baf 0  U03 
J = 0  d2 exp 0 k T 1 0 J :
(5)
@t
0
B
c0
This equation is easily solved under the condition J = Jc0 at t = 0 and we obtain!
kB T
2
Baf 0U03 t
J= 1 0 U 3 log  d2J k T + 1 :
(6)
Jc0
0
c0 B
0
Therefore, after a sucient time, 1 in the argument of logarithm can be neglected.
In this case, the logarithmic relaxation rate is given byJ  k Td(7)
0 d log t J = UB 3 ;c00
and U03 is obtained from experimental result. In usual case of experiment of relaxationof magnetization, initial magnetization, M0, is used and in this case J=Jc0 ! M=M0
for Eq. (7).

According to the result investigated by Welch[2], the relation between apparent
pinning potential and true pinning potential is given for sinusoidal pinning potential
well asU03 = 1:65(kB T U02)1=3 :(8)Therefore, true pinning potential can be estimated from apparent pinning potential
observed from relaxation of magnetization.
II.2. Flux creep in sinusoidal washboard potential
Here we estimate current density dependence of activation energy, U (j ), forsinusoidal washboard potential as shown in Fig. 2, where j is normalized current
density given by j = J=Jc0. The washboard potential is assumed by
F (x) =

U0

(9)
2 sin kx 0 fx
as shown in Fig. 1. In the above k = 2=af and f = JBV is Lorentz force to the ux
bundle, where V is volume of ux bundle. The local equilibrium position of the ux
bundle is obtained by di erentiating of Eq. (9) and given by!
1
2
f
x = 0x = 0 cos01
:
(10)
0

k

U0 k

On the other hand, F (x) is maximum at x = x0. Therefore activation energy is
obtained by U = F (x0) 0 F (0x0) and is2! 31=2
!
!
U 4
2
f 25
2
f
2
f
= 10 U k
0 U k cos01 U k :
U0
0
0
0

(11)

If there is no thermal agitation, virtual critical state is achieved and U = 0 is obtained.
In this case, x0 is equal to 0 and 2f=U0k = 1 is satis ed. Therefore, J is equal tovirtual critical current density in the case of ux creep free, Jc0. Hence, following
relation is obtained,
2f ! = J  j:
(12)
U0k
Jc0
From this equation, Eq. (11) is written as
U (j ) = U0[(1 0 j 2)1=2 0 j cos01 j ]:(13)
From the relationship,J
U 0 ' U + U0 ;(14)
Jc0
Eq. (1) can be written by"# U j 
U (j ) 
E = Ba  exp 0
1 0 exp 0 0 :
(15)
f 0

kB T

kB T

Therefore, E -J characteristics can be calculated under the assumption of U0 and Jc0.

Attempt frequency of the ux bundle, 0, for ux creep is investigated[3] andgiven by J
0 = f c0 ;(16)
2af B
where  is a constant depended on the kind of pinning center and  ' 2 is obtained
for the case of point like pinning center [4],  = 4 is obtained for the case of large
normal pinning center [5]. f is ow resistivity and according to Bardeen-Stephen
model [6], f is related with n and given by
f =

B
:
Bc2 n

(17)

n (T ) =T
 (T ):
Tc n c(18)

n is approximately given by

II.3. TAFF(Thermally Activated Flux Flow)
If the second term of Eq. (14) is smaller than kBT , Eq. (1) is written asBaf 0U0 J  U0 
exp 0 k T ;E'
Jc0 kB T
B(19)

and is linear E -J characteristics. This condition is known as TAFF (Thermally
Activated Flux Flow). After enough relaxation at high temperature or high magnetic
eld, TAFF can be observed. Here if J is sucient small, U ! U0 is achieved andresistivity is given by
 U 
(20)
 = 0exp 0 0 ;kB T
where 0 is equal to Baf 0U0=Jc0kBT . From above result, it is found that theresistivity under ux creep does not reach zero due to nite electric eld at nite
temperature. On the contrary, there are many discussions[7, 8] that if resistivity
will be zero at J ! 0 which represents real superconducting state (R = 0) at nite
temperature based on theory of glass-liquid transition of ux lines.
III.

FLUX CREEP-FLOW MODEL

In this section, we assume pinning potential, U0, which is most important parameter in the phenomena of the ux creep. Then, the electric eld induced by ux
ow is explained. It is shown how to calculate E -J characteristics, critical current
density Jc(B ) and irreversibility eld Bi(T ) based on the ux creep- ow model.
III.1. Pinning potential, U0
Fig. 3 shows the relationship between pinning force density and displacementof ux line. Ub0 represents pinning potential for unit volume of ux line. Whenthe displacement u is small enough, pinning force density increases linearly, and

F
Jc0 B
(a)
0

αL
0

u

di

U0

(b)

Fig. 3. (a)average pinning force density vs. displacement of ux line. (b) pinningpotential for unit volume of ux line, Uc0.
phenomenon is reversible in this region. On the other hand, it changes to nonlinear
and irreversible characteristics, and pinning force density is nally saturated to Jc0B ,when u reaches to interaction distance, di. Therefore, Ub0 can be estimated from thepresent pinning force density-displacement characteristics of reversible region. HereUb0 is given byd2(21)U^0 = L i ;2where L is Labusch parameter. Virtual critical current density Jc0 of ux creep freecase is related with L and di as
Jc0 B = L di :(22)
On the other hand, di is given by[4]adi = f ;(23)

where  is a constant depended on the kind of pinning center as abovementioned.
Then we have pinning potential U0 of ux bundle asU0 = U^0 V;(24)
where V is the volume of ux bundle. V is estimated from the size of ux bundle inlongitudinal direction, L and transverse direction R given byL=R= C 1=2
44
L C 1=2
66
L

;

(25)

;

(26)

R

R

R

R

L

L
d

Fig. 4. Volume of ux bundle, V . (a) when longitudinal ux bundle size, L, issmaller than thickness of superconductor, d, V = LR2. (b) when L is larger than d,
V = dR2.
as shown in Fig. 4(a). In the above, C44 is the tilt modulus of ux lines and given by
C44 =

B2
:
0

(27)
On the other hand, C66 is the share modulus of ux lines and is largely depended
on the condition of ux line lattice. In the case of perfect triangular lattice in three
dimensional, C66 takes maximum value of C660 , and it is given by[9]B 2
Bc2B 
= 4 B 1 0 B ;
0 c2
c2

(28)
and decreases with disorder of ux line lattice, then reaches to zero when ux line
lattice is in melt condition.
0
C66

III.2. Number of ux lines in the ux bundle, g2
It is considered that transverse ux bundle size, R is the order of af of several
times larger except the case of very week pinning of superconductor. In the case ofvery strong pinning force, R is smaller than af . Although, the size is not smaller thanux line spacing, R is expressed as
R = gaf ;(29)
where g2( 1) is the number of ux lines in the ux bundle. Therefore, g2 is given
by Eqs. (26) and (29)
C66
g2 =
:
(30)
J Ba
c0

f

3

10

Bi−2223 Ag sheathed
Bi−2212 single grain
Bi−2212 single crystal
2

g

2

10

#5

1

10

0

10 0
10

1

2

10

10

3

10

2

ge

Fig. 5. g2 obtained from experimental result and ge2 predicted by elastic theory[11].
Solid line represents the relation of Eq. (33).
For the case of three dimensional perfect triangle ux line lattice, g2 takes maximum
value, ge2, and it is given by[10]0C66= J Ba :(31)
c0f
As abovementioned, the value of C66 is largely depended on the condition ofux line lattice and changes from maximum value of C660 to zero, and there is noge2

straightforward method to determine the ux bundle size. Therefore, it is assumed
based on \minimization of energy dissipation" that g2 is determined so as to achieve
maximum critical current density under ux creep[11]. According to the theory, g2
is given as" Ba  #4=3
5
k
T
B
f 0
g 2 = ge2
log
;
(32)
2Ue
Ec
where Ue is the pinning potential at g = ge and Ec is criteria of electric eld, respectively. From the relationship of Ue / ge, we haveg 2 / ge2=3 :(33)
In Fig. 5, the relationship between g2 and ge2 is shown. g2 is obtained from the resultof irreversibility eld as aforementioned and is quantitatively explained from Eq. (33).
When size of bulk superconductor is larger than L and R, the volume of ux
bundle is given by
V = LR2 ;(34)

and the pinning potential is given by0:835g 2kB Jc01=2
U =
:
0

(35)

 3=2 B 1=4

On the other hand, if thickness of superconductor d is smaller than L such as a thinlm, V is given by
V = dR2(36)
and U0 is given by
4:23g2kB Jc0d :
(37)
U0 =
B 1=2
III.3. Irreversibility eld
Before explanation of numerical calculation by the ux creep theory, it is explained that irreversibility is analytically given by U0 derived in above. First we
assumed that the second term in Eq. (15) is neglected. Irreversibility eld is givenby the eld at Jc = 0 with de ned by the electric eld criteria E = Ec. Therefore
from Eq. (15) we have, U 
Ec = Baf 0exp 0 0 :
(38)
kB T
If we assume the temperature and magnetic eld dependences of Jc0 as"

Jc0 = A 1 0

 T 2 #m

Tc


B 
B 01 1 0
Bc2(39)

where A, m, and  are pinning parameters. Eq. (39) is usually used in conventional
metal superconductor and known as scaling law of the critical current density. For
the case d > L of larger bulk superconductor, irreversibility eld Bi is given by
Bi(302 )=2=

 K 2 "1T T 2 #m 

10 T

c

1 0 BBi

c2



:

(40)

In the above, K1 is a constant depended on pinning force density as
0:835g2A1=2 :K1 = 3=2(41)
 log(Biaf 0=Ec )On the other hand, for the case d < L such as thin lm superconductor, Bi is givenby"  2 #m 
K2
T
Bi 
(302 )=2
Bi= T 10 T
1 0 B d;(42)cc2
where K2 is given by
4:23g2A :(43)
K2 = log(Bi af 0 =Ec)In practical case, log(Biaf 0=Ec) can safely be regarded as a constant from 14 to 20,
since it is very weakly depended on Bi.

If temperature is near critical temperature, Tc, Bi is far below upper criticaleld, Hc2 and (1 0 Bi=Bc2) in right hand side of Eq. (40) can be regarded as unity.
In addition (K1=T )2 can be approximately put as a constant (K1=Tc)2. Therefore,
the temperature dependence of Bi is given as"

Bi / 1 0 T 2 #n

Tc

i

(44)

;

where temperature index of irreversibility eld, ni is given by
2m :
(45)
ni =
302
For example, in the case of Y-123, it is known that the parameters are given as
m = 3=2 and = 1=2, since the dominant pinning center at high temperature is
known as Y-211 precipitates. Then we have ni = 3=2 and the temperature dependenceof Bi is predicted as Bi / [1 0 (T=Tc)2]3=2. This result is well known as reported by
Yeshurun and Malozemo at very early stage[12].
III.4. Flux creep- ow model
For calculation of E -J characteristics, current density is not always smaller thanJc0 . At J > Jc0 , electric eld E is induced by ux ow, and is given by
E = 0;
j  1;= f (J 0 Jc0); j > 1(46)
where f is given by Eq. (17). On the other hand, electric eld by ux creep, Ecr, isconsidered to remain and assumed as[13, 14]#" U j 
U (j ) 
1 0 exp 0 0 ; j  1;
E = Ba  exp 0
cr

f 0



kB T
 U kB T= Baf 0 1 0 exp 0 k T0 ;j > 1:(47)
B
Then, the total electric eld, E , is obtained from Ecr and E and is approximately
given byE = (Ecr2 + E 2 )1=2 :(48)
In the above, E is approached to Ecr at j  1 and to E at j  1.
In usual oxide superconductor, pinning force density is known to be largely
distributed. Here only parameter A in Eq. (39) is assumed to have following distribution:#"(log
A 0 logAm )2;(49)
f (A) = K exp 022
where Am is most probable value of A, K is normalizing constant and 2 is a parameter representing the degree of distribution width of pinning force density. Therefore,
total electric eld is given by
Z1
Ef (A)dA:(50)
E (J ) =0

III.5. Summary for numerical calculation
Here, numerical calculation of E -J characteristics using the ux creep- ow
model is summarized. The virtual critical current density in the case of ux creep
free case is assumed in Eqs. (39) and (49). Therefore, pinning parameters Am, , m,and  should be determined. Then, according to the size of superconductor d and L
which is given in Eq. (25), pinning potential, U0, is given by Eqs. (35) or (37).
Electric eld by ux creep, Ecr, can be calculated with U0 as shown in Eq. (47),where attempt frequency of the ux bundle, 0, is given by Eq. (16) and activationenergy, U , is given by Eq. (13). On the other hand, electric eld by ux ow, E ,can be calculated by Eq. (46) with Eq. (17). Then nally, total electric eld, E , isobtained by Eqs. (48) and (50).
IV.

RESULTS AND DISCUSSION

In this section, some experimental results in oxide superconductor and othersuperconductor is compared with theoretical prediction by the ux creep- ow modelfrom references.
IV.1. E -J characteristics
E -J characteristics of Bi-2223 silver sheathed tape was investigated by four
probe method and relaxation of magnetization by Kodama et al.[15]. Figure 6 shows
the obtained results of E -J characteristics at 70 K. Four probe method was used formeasurement at high electric eld region, while the relaxation of magnetization is
1.40T

−5

E [V/m]102

σ =0.035−10

10

T =70K
0.08T

0.02T
7

10

8

2

J [A/m ]10

Fig. 6. Comparison of E -J characteristics between experiment (symbol) and theory(lines) at 70 K for Bi-2223 silver sheathed tape. Experimental results are obtainedby four probe method at high electric eld and the relaxation of magnetization at
low electric eld.[15]Table 1. Parameters used for theoretical analysis of magnetic eld dependence of Jc
of Bi-2223 silver sheathed tape[19]Tc Bc2k (0) n (Tc)Amm
  g2 2[K] [T] [ m]
108.3 50
100 3.02109 4.0 0.87 2.0 2 1.0 0.18
used at low electric eld. It is found that electric eld for 9{10 orders of magnitude
can be explained by the theoretical prediction of the ux creep- ow model. E -J curvechanges from convex to concave according to decreasing electric eld at B = 0:08 T.
IV.2. Magnetic eld dependence of Jc
There are several references for comparison of magnetic eld dependence of Jc
for Bi-2212[16], Y-123[17, 18], Bi-2223[19]. These results of Jc were obtained fromnumerically calculated E -J characteristics by ux creep- ow model. That is, Jc isdetermined at criterion of electric eld, Ec, such as Ec = 1 2 1004 V/m for the caseof four probe method.
Figure 7 shows the comparison of Jc(B ) between experimental and theoreticalresults of Bi-2223 silver sheathed tape at various temperatures[19]. Table 1 lists theparameters used in the calculation of Fig. 7.
108

Jc(A/m2)

107

40 K

106
46 K
105
60 K
1040

50 K

B‖c
1

2

3

4

B(T)

Fig. 7. Comparison of magnetic eld dependence of Jc between experiment (symbol)
and theory (lines) at various temperatures for Bi-2223 silver sheathed tape.[19]IV.3. Irreversibility eld
Theoretical irreversibility eld is obtained at the magnetic eld where Jc is
decreasing to the criteria, 1Jc such as 1Jc = 1 2 107; 1 2 106 A/m2. An example
of comparison of irreversibility eld between experimental and theoretical results of
Y-123 bulk specimens are shown in Fig. 8.
The irreversibility eld is largely depended on the kinds of experimental methodwith di erent electric eld criterion Ec and 1Jc. Figure 9(a) and (b) show dependencies of irreversibility eld on electric- eld criterion and temperature at 1Jc = 1 2 107
A/m2, respectively. It is found that Bi is largely depended on Ec corresponding toexperiment 10
experiment 11
theory 10
theory 11

B i (T)

4

2

0

60

70
T (K)

80

90

Fig. 8. Comparison of irreversibility eld between experiment (symbol) and theory(lines) for two Y-123 bulk specimens.[18](b)(a)
four probe method
hysteresis curves
relaxation
imaginary AC susceptibility(f0=350Hz)
imaginary AC susceptibility(f0=35Hz)0

10four probe method
hysteresis curves
relaxation

Bi (T)Bi (T)

10

0

−1

10
10

−1

T =70.0K
10

−10

10

Ec (V/m)

−5

0.2

0.3

0.4

0.5 0.6 0.7 0.80.9 1
2
1−(T/Tc)

Fig. 9. Dependencies of irreversibility eld on (a) electric- eld criterion and (b)temperature at 1Jc = 1 2 107 A/m2.[20, 21]

Table 2. Fitting parameters according to Eq. (40) for the Bi- Hg-based samples.
sampleTc [K] k = K1=Tc m
Bi-2223 thin lm 89
1.2
1.4 0.5(Bi,Pb)-2223 tape 103
1.0
1.9 0.5
Hg-1223132
0.92 2.0 0.55
(Hg,Pb)-1223
132
1.2
2.1 0.37
the kinds of experiments. And agreement between experiment and theory is good forwide range of Ec.
IV.4. Analytical result of irreversibility eld
As mentioned in section III.3., irreversibility eld is analytically expressed by
Eq. (40). Group of Luders has investigated the usefulness of Eq. (40) for analysis
of irreversibility eld in Hg-1223 and Bi-2223 specimens[22, 23]. Table 2 lists Tc
and parameters for four investigated specimens. Figures 10 and 11 show tting
results of temperature dependence of irreversibility eld for Bi-2223 specimens andHg-12223 specimens, respectively. It is found that the agreement between theoreticaland experimental results is fairly good for wide range of temperature, since factor K1
is used for temperature dependence of Eq. (40) in this work, which term is usuallytreated as constant and neglected. Therefore, irreversibility eld can be explained by
Eq. (40) for the wide temperature range in 0:2 < T=Tc < 1.

Fig. 10. Temperature dependence of the irreversibility eld for the two Bi-based
samples. The dotted t curves correspond to Eq. (40). Fitting parameters are given
in Table 2.[22]Fig. 11. Temperature dependence of the irreversibility eld for the two Hg-based
samples. The dotted t curves correspond to Eq. (40). Fitting parameters are given
in Table 2.[22]IV.5. Non-oxide superconductor
The ux creep- ow model is also used for analysis for non-oxide superconductor.
In Fig. 12 the theoretical results by ux creep- ow model of Jc of MgB2 are comparedwith measured values. Values of the parameters used for the calculation are given
in Table 3. Jc was determined with Ec = 1:0 2 1009 V=m corresponding to the DC
magnetization measurement. It is found that the theoretical result agrees well withthe experimental result except in high eld region at low temperatures. That is, thetheoretical result is much higher than the experimental result in this area.
It is important and useful for foreseeing the potential of material for application
with using the ux creep- ow model. First the critical current density will be estimated when ne normal precipitates are successfully dispersed in MgB2. The virtual
critical current density in this case is given by [24]N D2 0 Hc2
Jc0 = p
4af B

!
B 
10  H ;
0 c2

(51)

where Np and D are density and size of normal particles, respectively,  is the coherence length. Here we assume that D = 0:1 m and the volume ratio of normal
particles is NpD3 = 0:15. Equation (51) can be regarded as a most probable valueof Jc0, and hence we have m = 3=2, = 1=2,  = 1. In addition, Am = 4:8 2 109,
2 = 0:03 and g 2 = 1:5 are assumed. Using these parameters the E -J characteristics can be calculated and the critical current density can be de ned using the same
criterion as in experiment. These results are shown in Fig. 13, in which the present
experimental data are also shown for comparison. It is found that a high Jc value
can be achieved even at relatively high temperatures.

Table 3. Parameters Usedin Numerical Calculation.
Am
2 m
 g2
3:0 2 108 0.03 2.0 0.3 1.0 1.5
10K
15K
20K
25K
30K
35K

8

2

Jc (A/m )

10

6

10

4

10

10

−1

0

10

10

1

B (T)

Fig. 12. Theoretically analyzed Jc (solid lines) compared with experimental results(symbols) at 10, 15, 20, 25, 30 and 35 K.
10

10

10K
15K
20K
25K
30K
35K

8

2

Jc (A/m )

10

6

10

4

10

0

10

B (T)

Fig. 13. Calculated Jc (lines) assumed e ect of normal precipitates compared withprevious theoretical results (symbols).

V.

CONCLUSION

In this chapter, ux creep phenomenon was explained and some experimental
results on E -J characteristics, magnetic eld dependence of Jc, temperature dependence of Bi and so on are compared with prediction based on ux creep- ow model.
In general, electromagnetic phenomena of superconductor are well described by ux
creep- ow model.
REFERENCES[1] P. W. Anderson and Y. B. Kim: Rev. Mod. Phys. 36 (1964) 39.
[2] D. O. Welch: IEEE Trans. Magn. 27 (1991) 1133.
[3] K. Yamafuji, T. Fujiyoshi, K. Toko and T. Matsushita: Physica C 159 (1989)
743.
[4] A. M. Campbell, H. Kupfer, R. Meier-Hirmer: Proc. Int. Symp. on Flux Pinningand Electromagnetic Properties in Superconductors, Fukuoka (1985) 54.
[5] T. Matsushita: Jpn. J. Appl. Phys. 20 (1981) 1995.
[6] J. Bardeen, M. J. Stephen: Phys. Rev. 140 (1965) A1197.
[7] D. S. Fisher, M. P. A. Fisher and D. A. Huse: Phys. Rev. B 43 (1991) 130.
[8] M. P. A. Fisher: Phys. Rev. Lett. 62 (1989) 1415.
[9] E. H. Brandt: Phys. Rev. B 34 (1986) 6514.
[10] T. Matsushita, T. Fujiyoshi, K. Toko, K. Yamafuji: Appl. Phys. Lett. 56 (1990)
2039.
[11] T. Matsushita: Physica C 217 (1993) 461.
[12] Y. Yeshurun and A. P. Malozemo : Phys. Rev. Lett. 60 (1988) 2202.
[13] T. Matsushita and N. Ihara: Proc. 7th Int. Workshop on Critical Currents inSuperconductors (World Scienti c, Singapore, 1994) 169.
[14] T. Matsushita, T. Tohdoh and N. Ihara: Physica C 259 (1996) 321.
[15] T. Kodama, M. Fukuda, S. Nishimura, E. S. Otabe, M. Kiuchi, T. Kiss, T.
Matsushita, K. Itoh: Physica C 378{381 (2002) 575.
[16] E. S. Otabe, K. Okamura, H. Wada, M. Kiuchi, T. Yasuda, S. Okayasu, T.
Matsushita: IEEE Trans. on Appl. Supercond. 13 (2003) 3083.
[17] T. Matsushita, D. Yoshimi, M. Migita, E. S. Otabe: Supercond. Sci. Technol.
14 (2001) 732.

[18] D. Yoshimi, M. Migita, E. S. Otabe, T. Matsushita: Physica C 378{381 (2002)
788.
[19] M. Kiuchi, K. Noguchi, T. Matsushita, T. Kato, T. Hikata, K. Sato: Physica C
278, (1997) 62.
[20] T. Matsushita, M. Fukuda, T. Kodama, E. S. Otabe, M. Kiuchi, T. Kiss, T.
Akune, N. Sakamoto, K. Itoh: Advances in Cryogenic Engineering, 48 (2002)
1193.
[21] M. Fukuda, T. Kodama, K. Shiraishi, S. Nishimura, E. S. Otabe, M. Kiuchi, T.
Kiss, T. Matsushita, K. Itoh: Physica C, 357{360 (2001) 586.
[22] G. Fuchs, K. A. Nenkov, A. Attenberger, K. Luders, M. Baenitz, C. Ecker, K.
Kajikawa, E. V. Antipov, H. R. Khan: Physica C 355 (2001) 299.
[23] P. Shilbe, I. Didshuns, K. Luders, G. Fuchs, M. Baenitz, K. A. Lokshin, D. A.
Pavlov, E. V. Antipov, H. R. Khan: Physca C 372{376 (2002) 1865.
[24] A. M. Campbell, J. E. Evetts: Adv. Phys. 21 (1972) 372.