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Physica C 213 (1993) 477-482 North-Holland

Irreversibility line and flux pinning properties in high-temperature superconductors T. Matsushita, A. Matsuda and K. Yanagi Department of Computer Science and Electronics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820, Japan

Received 21 April 1993

The current-voltage characteristics are analysed numerically at high temperatures in terms of the one-dimensional flux creep model. A melt-processed Bi-2212 material including fine non-superconducting particles is considered as an example. A virtual critical current density in the creep-free case is assumed to approach the experimentally observed value at low temperatures, and a theoretical expression based on the flux pinning by non-superconducting particles at high temperatures where the effect of flux creep is significant. The critical current density is determined using the electric field criterion and the irreversibility field is obtained using a suitable criterion. The obtained irreversibility field shows different temperature dependences between low- and high-temperature regions as observed experimentally. The scaling characteristics of pinning force density are also discussed.

1. Introduction

The flux pinning characteristic and critical current density in high-temperature superconductors are largely influenced by the thermally activated flux creep at high temperatures because o f the weak pinning strength o f HTSC. This effect is more significant in Bi-2212 materials. The phenomena caused by the flux creep, such as magnetic relaxation, are well described by the flux creep model [ 1 ]. It is shown in ref. [ 2 ] that the irreversibility field is also derived by analysing the model with the help o f the collective pinning theory [3] in estimating the pinning potential from the critical current density observed in the low-temperature region where the flux creep is not effective. In this paper, this analysis is carried out more concretely for a melt-processed Bi-2212 single grain specimen with relatively strong pinning interaction, whose irreversibility field was observed [ 4 ] in order to prove the validity o f the model. This specimen contained Bi-free non-superconducting particles of a mean size o f about 5 ttm and a volume fraction o f about 10%. The observed critical current density and theoretical one for non-superconducting particles are used for estimation o f the pinning potential. The irreversibility field obtained from numerical calcula-

tion is compared with the experimental result. The scaling behavior o f the pinning force density is also discussed.

2. Theoretical model

The case is treated, for simplicity, where the thermally activated flux motion takes place one-dimensionally along the x-axis. The induced electric field is given by

1 where z, is the oscillation frequency o f the flux bundle inside the pinning potential, a is the hopping distance o f the flux bundle and can be approximated by the fluxoid spacing, af, U and U' are activation energies on both sides o f the flux motion and kn is the Boltzmann constant. The oscillation frequency is given by [ 5 ] Of Jco U= 2~arB'

(2)

where ( is a constant dependent on the pinning center and is believed to take 2n for pointlike defects

0921-4534/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

478

T. Matsushita et aL / Irreversibility line and flux pinning properties

[ 6 ], pf is the flux flow resistivity approximately given by (B/Bc2)pn with Be2 and p, denoting the upper critical field and the normal resistivity, respectively, and Jco is the virtual critical current density in the creep-free case. If we assume the sinusoidal washboard potential, the activation energy of the side directed to the Lorentz force is given by [ 7 ] U= Up[ ( 1 _j2) l / 2 j COS-Ij] ,

(3)

wherej=J/Jco is the reduced current density and Up is the pinning potential. In this case eq. ( 1 ) is written as

E = B a f v e x p ( - ~ T ) [ 1 - e x p ( - n U ° j ~ kBT ] ]_]"

(4)

The collective pinning theory [ 3 ] predicts the pinning potential as a function of the virtual critical current density as

Up 0.835gZJlc~2 kB -- ~3/2B1/4 ,

(5)

where g2 is the number of fluxoids in the flux bundle. Thus, if Jco is given, the current-voltage characteristics are simply obtained from eqs. ( 2 ) - ( 5 ). In this paper the numerical analysis is done for a melt-processed Bi-2212 single grain specimen with non-superconducting particles with the magnetic field normal to the c-axis [ 4 ]. The observed irreversibility field had different temperature dependences between low- and high-temperature regions. This was ascribed to different pinning centers. The assumed superconducting and pinning parameters are listed in table 1. Although the ff value assumed here is applicable for pointlike defects which are believed to be effective at low temperatures, this value is approximately used also for non-superconducting particles effective at high temperatures. The g2 value estimated on the irreversibility line in the temperature Table 1 Superconducting and pinning parameters of a melt-processed Bi2212 single grain specimen assumed for numerical analysis

Tc Bc2(0)

92.8 K 690T

Pn g2

1.0X 10 -4 2.0 2n

T/Tcf~m

region below 70 K was 2.2 [4]. We neglect a slight variation in g2 with temperature and magnetic field and, for simplicity, assume g2= 2 in the entire regions of temperature and field. The virtual critical current density at low temperatures is expected to vary with temperature and field as J I = 7 . 5 2 X 10s[1--(~c)21225B-°'21(1--~c2) 2. (6) The responsible pinning centers are considered to be pointlike defects. On the other hand, non-superconducting particles of the Bi-free phases are expected to be effective at high temperatures. We assume this contribution as [[ T ~2-]3/2 [ B 2 (7)

:~:3.33xlo,[1-L~)]

B-'/E~,I-~-.j-2) .

Although this contribution is believed to be slightly smaller than the theoretical estimate, ,/2(0 K, 1 T) = 7.67)< l0 s A / m 2 [4], for the case of the mean size and volume fraction of the particles of 5.0 pm and 10%, respectively, this seems to be reasonable because of the use of a greater g2 value than the theoretical estimate, g2= 1. The actual critical current density is considered to have the above two components. The collective pinning theory suggests that the resultant virtual critical current density is given in the form

Jco = (J,~ + j ~ ) , / 2 .

(8)

The critical current density is defined by the electric field criterion of E = Ec = 1.0 )< 10- 5V/ m. Because of this criterion and the finite flux-flow resistivity, a certain critical current density remains even in the ohmic case. Hence, the irreversibility field is defined as the field where the critical current density decreases to 1.0)< 105 A / m 2.

3. Results and discussion

Figure 1 shows an example of the current-voltage characteristics at 70 K and 1.0 T. It is seen that the characteristic is ohmic at very low current density, showing the thermally assisted flux flow state, and increases nonlinearly with increasing current den-

T. Matsushita et aL / Irreversibility line and flux pinning properties -

479

This is attributed to different pinning characteristics between the two temperature regions. In the previous calculation [ 2 ], where a single pinning characteristic was assumed, the resultant irreversibility field showed a single n value. The straight lines represent the temperature dependences in low- and high-temperature regions predicted by the theory. That is, if the scaling relation o f ,/co in the low-field region is given by

10-3 70K, 1.0T 10-4 E 10-5 10-6

\ ~ - j j B:'-', 10-7

,t

106

i

i

i

i

,i,,I

107 J(A/m2)

108

Fig. 1. Calculated current-voltage characteristics at 70 K and 1.0 T.

T(K) 90

"~.z5~

10° n=l

~

10-2

,,

~ n=3.17 , 10-1 1- (T/Tc)2

where A is a constant and the upper critical field is assumed to vary as B ¢ 2 ( T ) = B ¢ 2 ( O ) [ 1 - ( T / T ¢ ) 2 ] , we have 2m n = 3_2-~

(11)

in the vicinity o f the critical temperature. At low temperatures n is more influenced by J~ and n = 3.17 is expected. The deviation from the straight line at much lower temperatures comes from the approximation o f ( K / T ) 4/(3-ay) by the constant (KITe) 4/(3-2~) in eq. (A.2) in the Appendix. The dotted line in the figure gives the exact analytical prediction. On the other hand, n = 1.5 is expected from J2 at high temperatures. This numerically calculated result is compared with the experimental result [4] shown in fig. 3. The two temperature de-

80 70 50

101

10-1

(10)

100

T(K)

Fig. 2. Calculated irreversibility line corresponding to the experimental result shown in fig. 3. The two straight lines represent the theoretical predictions given by eq. ( 11 ) at low-and hightemperature regions. The dotted line is the correct theoretical prediction without approximation.

9O

8O 70 50

r

101

,

3

i

T

~

p__. lO o

sity. Figure 2 is the obtained irreversibility field, which is plotted in the form o f

B i ocI 1

" T-2~", - ('~) J

//n183 n=1.7

n=3.83

(9,

as predicted by the collective flux pinning theory [ 3 ]. The theoretical results are described in the Appendix. The temperature dependence is found to be different between low- and high-temperature regions.

10-2

10-1 1-(T/Tc)2

.... 100

Fig. 3. Experimental result of irreversibility line in a melt-processed Bi-2212 single grain specimen [4 ].

T. Matsushita et al. / Irreversibility line and flux pinning properties

480

pendences in the low- and high-temperature regions shown in the calculated result agree fairly well with those, i.e., n=3.83 and n = 1.73, in the observed result. The larger n at low temperatures seems to be attributed to the above approximation and the smaller n at high temperatures may be due to a lack of observation in the high-temperature region of 1 - (T/To) z < 0.1. The quantitative agreement is also fairly good. The difference seems to be attributable to various approximations in the calculation. Hence, it is concluded that the flux creep theory correctly explains the irreversibility line, suggesting that the irreversibility line is determined by the depinning of fluxoids due to the thermal activation. Next we shall discuss the pinning properties in the vicinity of the irreversibility line. The pinning force density is obtained from the critical current density as Fp=JcB. If the pinning force density at a given temperature is normalized by its maximum value, Fpm, and plotted as a function of the reduced magnetic field, b=B/Bi, the scaling relation is obtained as shown in fig. 4. At high temperatures, the normalized pinning force density takes relatively large values because of the use of a finite criterion for the irreversibility line. This results in a slight deviation from the universal curve with a shorter high-field tail at high temperatures. If the pinning force density is approximated by a well-known formula given by

Fp=A'bP(l-b)q,

(12)

the parameters A', p and q at various temperatures are given in table 2. The q value is much larger than

50K 60K ,, 70K 80K • 90K •

'\ ,\

- V\

~0.5

~i "~r

Vo

0.5

1

B/B i

Fig. 4. Normalized pinning force density vs. magnetic field at various temperatures.

Table 2 Scaling parameters

of pinning force density at

various

temperatures T(K)

A'

p

q

50 60 70 80

1.78X 109 5.15X 108 1.36X 108 2.2l X 107

0,70 0,62 0.58 0.47

3.52 3.41 3.24 2.83

T(K) 70

80

60 50

/

0.~T 108 Jco ~"

•

-~

m=3.45

10 7 /-

~o 10 6

10 5 100 1 - (T/Tc) 2 Fig. 5. Temperature dependence of critical current density at 0.6 T. The virtual value J¢o is also shown for comparison. The m value estimated from the region between 50 and 70 K is 3.45.

2, which is observed for Nb3Sn [8 ] or Y - B a - C u - O [ 9 ]. This is considered to be caused by the larger influence of the flux creep than in materials with stronger pinning forces. The influence of the flux creep on the pinning property in the Bi-2212 superconductor is now discussed quantitatively. Figure 5 shows the temperature dependence of the critical current density at 0.6 T. The virtual critical current density Jco is also shown for comparison. It is seen that the degradation with increasing temperature is much more significant than that of Jco. Hence, it is understandable that m cannot be exactly determined from observed results at such high temperatures. For example, if rn is determined in the temperature region between 50 and 70 K, m=3.45 is obtained. However, this value is much larger than the actual value, 2.25. In fig. 6 the virtual and resultant pinning force densities are compared at 70 K. Because of the drastic decrease of the ir-

T. Matsushita et aL / lrreversibifity line and flux pinning properties

.....

10 9

i

. . . . . . . .

I

.....

I

. . . . . . . .

70K

i

,

r

~~/

"

Fp 0

/

108

107

.....

10 6

10_1

10 0

I

101

. . . . . . . .

I

10 2 '

481

comparison with that observed in Y-123 superconductors. This seems to be ascribable to weaker pinning strength in the Bi-2212 superconductor. (3) The influence of the flux creep on the pinning properly was clarified quantitatively. For example, the irreversibility field was reduced to below 1% of the upper critical field and the maximum pinning force density was about 3% of the virtual maximum value at 70 K.

Acknowledgement

B(T) Fig. 6. Comparison o f actual and virtual pinning force densities at 70 K.

reversibility field from the upper critical field, it can be concluded that the pinning force density is significantly depressed by the flux creep at these temperatures. It is necessary, therefore, to increase the pinning strength in order to improve the pinning property in Bi-2212 superconductors. This numerical analysis is useful for foreseeing the pinning property of strongly pinned superconductors which will be realized in the future.

4. Conclusions In this paper numerical calculations of the irreversibility field and pinning force density were carried out using the flux creep model on a melt-processed Bi-2212 single grain which contains non-superconducting particles. The results were compared with experimental results and the following conclusions were obtained. (1) The characteristic irreversibility field with different temperature dependences between low- and high-temperature regions observed in the experiment was reproduced theoretically. The obtained irreversibility field also agrees quantitatively with experiments. This shows that the flux creep model correctly explains the irreversibility line, suggesting that the irreversibility line is the depinning line. (2) The scaling behavior of the pinning force density was found to be as usually observed. The highfield tail of the pinning force density was longer in

This study is supported by the Grant-in-Aid for Scientific Research on Priority Areas, "Science of High Tc Superconductivity" given by the Ministry of Education, Science and Culture, Japan.

Appendix According to the flux creep model, the irreversibility line is determined by the condition at which the critical current density defined by the electric field criterion E=Ec is reduced to zero. Since the activation energy approaches the pinning potential, Uo, in this limit, this requirement is written as {

Uo =kB T l n ~ - - ~ - ) .

(A.1)

In the above the second term in eq. ( 1 ) is approximately neglected. Substitution of eqs. ( 5 ) and (10) into eq. (A. 1 ) derives

(T

m''32" /

,

(g.2)

where the constant K is given by

K= 0"835gE[AB~2(O) ]1/2 ~3/Zln ( Bafl~/ Ec )

( A. 3 )

Hence, in the vicinity of the critical temperature, the factor of ( K / T ) 4/~3- zy) is approximately replaced by (K/Tc) 4/(3-27) and eqs. (9) and (11 ) are obtained. The deviation from these equations becomes appreciable with the larger B i value at lower temperatures.

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T. Matsushita et al. / lrreversibility line and flux pinning properties

References [1] P.W. Anderson and Y.B. Kim, Rev. Mod. Phys. 36 (1964) 39. [2] T. Matsushita and A. Matsuda, Adv. Superconductivity V (Springer, Tokyo, 1993 ) p. 463. [3] T. Matushita, T. Fujiyoshi, K. Toko and K. Yamafuji, Appl. Phys. Lett. 56 (1990) 2039. [ 4 ] T. Matsushita, T. Nakatani, E.S. Otabe, B. Ni, T. U memura, K. Egawa, S. Kinouchi, A. Nozaki and S. Utsunomiya, Cryogenics 33 ( 1993 ) 251.

[5] K. Yamafuji, T. Fujiyoshi, K. Toko and T. Matsushita, Physica C 159 (1989) 743. [6] A.M. Campbell, H. Kiipfer and R. Meier-Hirmer, Proc. Int. Symp. on Flux Pinning and Electromagnetic Properties in Superconductors (Matsukuma, Fukuoka, 1985) p. 54. [ 7 ] T. Matsushita and E.S. Otabe, Jpn. J. Appl. Phys. 31 ( 1992 ) L33. [ 8 ] E.J. Kramer, J. Appl. Phys. 44 ( 1973 ) 1360. [9] T. Nishizaki, T. Aomine, I. Fujii, K. Yamamoto, S. Yoshii, T. Terashima and Y. Bando, Physica C 181 ( 1991 ) 223.