Experimental studies on magnetic relaxations in YBa2Cu3O7-δ: logarithmic, power law or collective pinning?

Experimental studies on magnetic relaxations in YBa2Cu307_ " logarithmic, power law or collective pinning?* Y. Ren* and P. A. J. de Groot Department of Physics, Southampton University, Southampton SO9 5NH, UK Magnetic relaxations have been measured for up to 20 h for a flux-grown Y B a 2 C u 3 0 7 _ ~ (YBCO) single crystal, with the applied field ( 1 - 8 T ) parallel to the c-axis, for temperatures from 10 to 70 K. Results from the experiments are compared with the theoretical models. It has been found that the relation from the collective pinning theory, M(t) = M o [ 1 + (l~kT/Uo) In (t/r o)] (- 1/M gives an overall description of the experiments. The logarithmic relation ( A n d e r s o n - K i m model) and the power law relation, M oc t ~, are simply approximations of the collective pinning theory under certain conditions: logarithmic for low temperatures and fields; power law for both low and medium temperatures and fields. Values of the exponent/~ in the collective pinning relation are also given, as obtained from the experiments.

Keywords: flux creep; magnetic relaxation; Y B a 2 C u 3 0 7 _ ~

Magnetic relaxations have been observed in type II superconductors, including high T~ oxides ~ 3. The origin of this phenomenon has been interpreted by a thermally activated flux creep theory, first proposed by Anderson and Kim ( A n d e r s o n - K i m model) 4'5. It is assumed that uncorrelated flux lines or bundles of flux lines can be thermally activated and jump over pinning barriers. The hopping rate u can be described by the Arrhenius law

= ~oexp

I

kT

J

(1)

where u0 is an attempting frequency and U(J) is a current-dependent effective energy barrier. It is assumed that U(J) depends on the current density J linearly

U(J)=

Uo 1 - Jo

(2)

where U0 is the J = 0 barrier and J0 is the critical current density in the absence of thermal activation. This, * Paper presented at the conference 'Critical Currents in High To; Superconductors', 2 2 - 2 4 April 1992, Vienna, Austria + Present address: Institute for Beam Particle Dynamics and Texas Center for Superconductivity, University of Houston, Houston, TX 77204-5506, USA

together with the assumption kT,~, Uo, leads to a logarithmic relaxation of the magnetization M(t) = M°[ 1 - kTUoI n ( ; 0 ) ]

(3)

where r0 is a time constant and 1/70 is related to the attempting frequency u0 for the flux jumps. Although it is not clear if these approximations are appropriate, Equation (3) is extensively used for high Tc superconductors. Recent studies, however, have shown that departures from the logarithmic relation occur in certain temperature and field regions, for different time intervals ranging from less than 3 h to 4 days 6 ~. Alternative models have been suggested to interpret the non-logarithmic time decay of the magnetization. Zeldov et al. 9 have found that their experimental results on the resistive transitions in strong magnetic fields with different current densities can be readily described assuming a logarithmic dependence for U(J), i.e. U(J) = Uo ln(J0/J)

(4)

Maley et al. ~o have also found the above relation from analysis of their magnetization data. Vinokur et al.l~ have shown theoretically that Equation (4) leads to a

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Magnetic relaxations in YBCO: Y. Ren and P.A.J. de Groot

power law relaxation. In the case of full flux penetration

M ( t ) = M o ( t ~ -~ \r0/

(5)

where ot = kT/Uo and 70 is an effective time constant. A possible explanation of the non-logarithmic relaxation emerges from recent theories involving collective pinning 12't3. This theory predicts an 'interpolation relation'

[

,,

M(t) = M0 1 + /~U001n

(6)

Phenomenologically this implies an effective barrier

Using a different approach (usually referred to as the vortex glass theory), Fisher et al. 14,t5 have given the same prediction. In the following, we show our magnetic relaxation data obtained over a wide temperature and field range for a flux-grown YBCO single crystal. We compare our experimental results with the above-mentioned theoretical models. We have found that the collective pinning-vortex glass theory gives an overall explanation of our data. On the other hand, both the logarithmic and the power law relations can be treated as approximations of the collective pinning model.

Experimental procedures The YBCO single crystal used in this experiment, supplied by the Birmingham University crystal growth facility, was grown from a NaCI:KC1 flux, using prereacted and fine-ground YBCO powders as the starting material 16-18. Such crystals are found to be heavily twinned and contain many defects, providing enough flux pinning centres to support a relatively high critical current density. The sample has a mass of 54.98 mg and a size of = 3 . 3 × 2.4 x 1.1 mm 3, with the c-axis of the crystal lattice corresponding to the shortest length. A.c. susceptibility measurements show an onset Tc of 92 K. Typical values for critical current densities in the a-b plane, with the applied field Bile, are Jc = 1.6 × 108 A m -2 for 70 K and 1 T, and Jc = 2.7 X 10~A m -~ at 20 K in the same field, as extracted from the widths of the magnetic hysteresis via the Bean model t9. Magnetization measurements were carried out on an Oxford Instruments vibrating sample magnetometer (Model 3001). The sample was first zero-field cooled to a desired temperature and a field, parallel to the c-axis, then applied. The values of the applied field were chosen to ensure full penetration of the field to the centre of the sample. The time variation of the magnetization M(t) was recorded at an interval of 100 s for 2 0 h (7.2 × 10 4 S). However, the data taken in the first 5 min (3 x 102 s) were excluded in later analysis as they

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might be affected by transients in the measuring system. After each measurement was completed, the sample was warmed up to 100 K for removal of the trapped flux lines inside the sample.

Results and analysis We made two series of measurements: 1, for a constant applied field of 2 T with temperatures from 10 to 70 K (the '2 T series'); and 2, for a constant temperature of 50 K, in applied fields from 1 to 8 T (the '50 K series'). In Figure I we show examples of these measurements plotted as M(t) versus In(t). We found for the field of 2 T and a relatively low temperature of 10 K that the magnetization exhibits a linear relation with In(t). For temperatures of 20 K and above, deviations from the linear relation were observed. The departure from the linear relation is more obvious for higher temperatures, especially above 50 K. For the 50 K series, none of the relaxations has a good logarithmic time dependence, especially for relatively higher fields. To examine the power law relation, we made doublelogarithmic plots for our data. We found for the 2 T series that the plots are linear from 10 up to 50 K, but start to curve slightly at 60 K and are more curved for 70 K. For the 50 K series the plots are linear for fields from 1 to 3 T, start to bend a little at 4 T and become more curved at 7 T. A least-squares fit shows that for a temperature of 10 K and a field of 2 T, both the logarithmic and the power law relations describe our data equally well; while for intermediate temperatures and fields, the power law relation gives a much better fit (Figure lb). For relatively high temperatures and fields, however, neither the logarithmic nor the power law relations give good fits (Figure lc). For the entire temperature and field range investigated, our experimental data are described very well by the collective pinning theory [Equation (6)], which we rewrite as

M(t) = A [B + In t] - l/,

(8)

with A, B and /z as fitting parameters. Especially for relatively high temperatures and fields, only the collective pinning model gives a good fit (see, for example, Figure lc). The/z values thus determined are given in Figure 2. An alternative method of extracting /~ is to replot the data as M -~ versus In(t). The # value is then extracted from the criterion of the best straight line (Figure 3). The/z values found by both methods are in good agreement.

Discussion From the above analysis, we found that the relaxation behaviour of the YBCO single crystal is closely related to the existence of the irreversibility line. The A n d e r s o n - K i m model is valid only for temperatures and fields far below the irreversibility line. On the other hand, the power law relation is satisfied for a wide region in the B(T) plane. As the temperatures and the fields increase towards the irreversibility line, the power law relation also breaks down; and here only the collective pinning model still gives a suitable description.

Magnetic relaxations in YBCO: Y. Ren and P.A.J. de Groot

-1.75

2.5

-1.80

(a)

E

~" -1.85

2

(a)

1.5

CD

"' -1.90-

1

-1.95

0.5 6.0 7)0 8'.0 9'.0 16.0 li.0 12.0 In(t)

00

2'0 40 60 80 Temperature (K)

100

2.5

-0.15-

_

E

-0.20-

1.5-

(b)

ID

1

-0.250.5

S

-0.3~1.0 6.0 7LO 8'.0 9'.0 10.0 li.O 12.0 In(t)

00

4

6

10

Applied Field (T) Figure 2 (a) Temperature dependence of /z for YBCO single crystal in applied field of 2 T parallel to the c-axis. (b) Field dependence of # for the same sample at a t e m p e r a t u r e of 50 K

-0.002~'-

-0.004-

'~

-0.006

E

-0.008 r

-0"0~1.0 6'.0 7'.0 8'.0 9'.0 10.0 li.0 12.0 In(t) Figure 1 Examples of magnetization v e r s u s In(t) plots for YBCO single crystal, with BIIc. (a) For low temperature and field (T = 10 K, B = 2 T) the logarithmic relation gives a good description of the experimental results. (b) For intermediate t e m p e r a t u r e and field (T = 50 K, B = 2 T), deviation f r o m the logarithmic relation is obvious. The solid line is a least-squares fit to the p o w e r law relation M(t) = At % with c~ = 0 . 0 8 9 and A = - 0 . 5 0 × 106 A m 1. (c) For the same temperature (50 K) but in a higher field of 7 T, both the logarithmic and the p o w e r law (dotted line) relations fail to describe the data. The solid line is a fittoM =A[B + I n ( t ) ] - ( l / " l , w i t h # = 0.6, A = - 0 . 2 3 x 1 0 6 A m 1 and B = 1.91

Simple mathematics show that the A n d e r s o n - K i m equation [Equation (3)] can be treated as an approximation of the collective pinning equation [ Equation (6) ] if #(kT/Uo) In(t/r0),~ 1. The same logarithmic relation can also be derived by taking the first-order terms in the series expansion of the power law relation [Equation (5)] when o~ = kT/Uo is small. Indeed we found that all three models fit our measurements at 10 K in a 2 T field with almost equal accuracy. The power law relaxation can also be derived from Equation (6) using a series expansion, provided that ~ is small. Using the temperatures and applied fields for which our experimental data obey the power law relation, almost equally good or even better fits can be achieved for the collective pinning model. The/x values obtained for this temperature and field region fall in the range 0.2 < / z < 0.5. We see that the collective pinning model gives a general description of the relaxation phenomenon. The two other models, namely the logarithmic and the power law relations, are simply its approximations under different conditions. From the vortex glass theory, Fisher et al.]5 have predicted /x to be a universal exponent less than 1.

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Magnetic relaxations in YBCO: Y. Ren and P.A.J. de Groot

.,-t

60 55 50

%. 45 ~' 4035 30

2~d.0 6.0 710 810 910 10.0 li.0 12.0 In(t)

10-

_ [..,

6'

m

4.

%

6'0

Feigel'man et al. 12 found that for collective pinning, # depends on the flux bundle size. In the threedimensional case, # = 1/7 for individual vortex hopping, /~ = 3/2 for small bundles and /z = 7/9 for large vortex bundles. For hopping distances comparable to the flux lattice constant, Nattermann a° has given # = 1/2. Our experimental results show that for low to medium temperatures and fields, /z is almost independent of temperature and field. However, at a certain threshold it rises sharply (see Figure 2). It is interesting to notice that Huang et al. 7 have found sharp changes in the logarithmic relaxation rate R = d l n ( M ) / d l n ( t ) at similar threshold B(T) values, which they believe correspond to a possible new phase transition. Figure 4 is a reproduction of Figure 7 in reference 7, on which we have superimposed the B(T) points where the/~ values obtained in our experiments have sharp changes. The /z values so far reported are controversial. For example, Thompson et al. 8 found from their magnetic relaxation measurements for an YBCO single crystal that the value of/z lies between 0.5 and 2, and that it first increases with the temperature and the field, then passes through a maximum, and afterwards decreases for still higher temperatures and fields. On the contrary, from transport measurements for a thin film sample, Dekker et al. 22 found/x = 0.94 for low temperatures and fields; a value which fell to 0.19 at high temperatures and fields. /z values as high as 2.5 have also been reported for melt-textured samples 23. We should point out that these experiments involve different measuring techniques and different samples. Since the pinning mechanism is closely related to the microstructure of the sample, the exponent /z must be sample dependent. Therefore the above-mentioned values may not be directly comparable. Another difficulty in obtaining/x values is that/x is not a directly measurable quantity. All /z values are extracted from curve fitting, no matter what techniques are involved in the experiments. It has been found that there is a strong correlation between the fitting parameters (in our case, A, B and /z) 23'24. In such a situation small systematic errors in the data may lead to large deviations in the fitting parameters, and a change in one parameter will cause deviations in the others.

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100

T (K)

Figure 3 Magnetic relaxation of YBCO single crystal, for 50 K and 7 T, plotted as M - ~ versus In(t), with/~ = 0.6 (same set of data as in Figure lc)

80

Figure 4

Reproduction of Figure 7 in reference 7. The solid line represents the irreversibility line. • , • , Possible phase boundary measured for t w o samples (reference 7); +, B(T) points at which # changes sharply (this work)

Further studies are needed to understand the discrepancies between these experiments. For example, it is worth examining the same sample with different measuring techniques, and examining different samples with the same measuring method.

Summary In summary, we have measured the magnetic relaxation of an YBCO single crystal over a 20 h time period. We have found that Anderson-Kim logarithmic relaxation is satisfied only for low temperatures and low fields, simply because it is just the first-order approximation of the more general cases of either a power law relation or a relation obtained from the collective pinning model. The power law relation (which is the approximation of the collective pinning model when # is small) is valid for a wide range of temperatures and applied fields. For higher temperatures and fields relatively close to the irreversibility line, both the Anderson-Kim model and the power law relation fail to describe the experimental data, while the collective pinning theory can still give an explanation for the magnetic relaxation.

Acknowledgements The YBCO single crystal used in this study was kindly supplied by the University of Birmingham crystal growth facility. This research is supported by the UK Science and Engineering Research Council.

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Magnetic relaxations in YBCO: Y. Ren and P.A.J. de Groot 7 Huang, Z.J., Xue, Y.Y., Hor, P.H. and Chu, C.W. Physica C (1991) 176 195 8 Thompson, J.R., Sun, Y.R. and Holtzberg, F. Phys Rev B (1991) 44 458 9 Zeldov, E., Amer, N.M., Koren, G., Gupta, A. et al. Phys Rev Lett (1989) 62 3039 10 Maley, M.P., Wills, J.O., Lessure, H. and McHenry, M.E. Phys Rev Lett (1990) 42 2639 11 Vinokur, V.M., Feigel'man, M.V. and Geshkenbein, V.B. Phys Rev Lett (1991) 67 915 12 Feigel'man, M.V., Geshkenbein, V.B., Larkin, A.I. and Vinokur, V.M. Phys Rev Lett (1989) 63 2303 13 Feigel'man, M.V., Geshkenbein, V.B. and Vinokur, V.M. Phys Rev B (1991) 43 6263 14 Fisher, M.P.A. Phys Rev Lett (1989) 62 1415 15 Fisher, D.S., Fisher, M.P.A. and Huse, D.A. Phys Rev B (19911 43 130

16 Abell, J.S., Darlington, C.N.W., Drake, A., Hollin, C.A. et al. Physica C (1989) 162/164 909 17 Drake, A., Abell, J.S. and Sutton, S.D. J Less Common Metals (1990) 164/165 187 18 Gencer, F. and AbeU, J.S. J Crystal Growth (1991) 112 337 19 Ren, Y., de Groot, P.A.J., Gencer, F. and Abell, J.S. Supercond Sci Technol in press 20 Nattermann, T. Phys Rev Lett (1990) 64 2454 21 Dekker, C., Eidelloth, W. and Koch, R.H. Phys Rev Lett (1992) 68 3347 22 Xue, Y.Y., Gao, L., Huang, Z.J., Hor, P.H. et M. Proc 1992 TCSUH Workshop on HTS Materials, Bulk Processing and Bulk Applications Houston, Texas, USA (Feb 1992) in press 23 Xue, Y.Y., Gao, L., Ren, Y.T., Chan, W.C. et al. Phys Rev B (1991) 44 1209 24 Sandvold, E. and Rossel, C. Physica C (1992) 190 309

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