Non-Debye relaxation and dynamical scaling in high-Tc granular superconductors

Physica A 168 (1990) 277-282 North-Holland

NON-DEBYE RELAXATION AND DYNAMICAL SCALING IN H I G H - T c GRANULAR SUPERCONDUCTORS A. G I A N E L L I and C. G I O V A N N E L L A Dipartimento di Fisica, Sez. INFN and Consorzio INFM, Universitft di Tor Vergata, Via E. Carnevale, 00173 Roma, Italia

An analysis is presented of the dynamical behaviour of a sintered pellet of YBCO, in terms of a general model for relaxation in complex systems. A nice similarity is put in evidence between the critical dynamics of the high-Tc granular superconductors and the spin-glass one.

1. Introduction

The first investigations of granular superconductors as disordered and critical ensembles of Josephson junctions date to the late sixties and the early seventies [1-4] but only recently, because of the discorvery of the high-T c superconductors [5, 6], the interest in this subject has strongly renewed and increased. As is now well known, in these sintered materials there exist at least two nested networks of junctions (either Josephson or weak link type): an intragranular and an intergranular one; our interest, since the beginning, has been focused on the study of the phenomenology of the latter [7, 8]. The static properties of these junction networks in the proximity of their critical transition have been studied by means of a S Q U I D magnetometer in very low magnetic fields, and can be interpreted by making use of the standard scaling hypothesis, within the framework of the fractal cluster model [9, 10]. Moreover, the values of the critical exponents suggest a possible analogy between spin glasses and granular superconductors [9-11], made more stringent by the results of computer simulations that evidenced the relevance of the frustration [12-14] in determining the energy and the configuration of, at least, the ground state of a disordered Josephson junction network [15]. In this paper we present, and analyze, data on the critical dynamics of sintered high-T c superconductors. The present study of the dynamical properties of a sintered pellet of Y B C O has been pursued by means of a standard ac magnetic susceptometer and a flux exclusion probe [16]. We started this investigation hoping to derive (exactly as has been done for spin glasses [17]) from the non-linear response all the critical exponents characteristic of the transition. However, although the observed 0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)

278

A. Giannelli and C. Giovannella / Dynamics in high-T c granular superconductors

phenomenology corresponded to what we expected, the existence of the two nested levels of junctions introduced unexpected difficulties in the data analysis [18], and thus required a more careful analysis of the linear response, in order to understand in which limits we are allowed to apply the standard scaling hypothesis, and to which extent the dynamics of the granular superconductors is similar to that observed in other glassy systems. As is well known, the shape of the ac susceptibility curves (both the real and the imaginary components) is affected by the intensity of the ac probing field and by that of an eventual superimposed dc field, as well as by the frequency of the ac field [19, 16]. The critical dynamics of quasi-equilibrium can be studied only performing the measurements in very low dc fields and, that is even more important, using a very low ac field. The modifications of the shape of the susceptibility curves induced by intense dc and ac fields belong to the domain of the non-equilibrium dynamics and have been explained by several researchers within the framework of either a modified Bean model or a Kim model [19]. None of these models, however, is able to explain the observed frequency effect.

2. Non-Debye relaxation In fig. 1 we show a series of X' versus X" plots for three different measuring frequencies. We note immediately that the three curves are all extremely skewed, and that they can be fitted by a Cole-Davidson law (X * = [1/(1 + itoT) ~l-n)] - 1 ) but only up to a critical temperature marked by the star. A combination of the C o l e - D a v i d s o n and the C o l e - C o l e laws (X * = {1/[1 + (ito~-)l-~] (l-n)} - 1) does not improve the fit. These phenomenological laws are usually used (for a very large ensemble of disordered materials [20]) to fit susceptibility data taken at constant temperature as a function of the probing frequency. Here we use a constant frequency and vary the temperature, which is the same as varying zp if the dynamics is controlled by an activated process of the form % = T Oexp[U(T)/kT]. This is a correct procedure if one considers that the plots of fig. 1 depend only on the product to~-p. Zp may vary either because of the change in T or because of an eventual change in U(T). Indeed we observed either a temperature region in which U is basically constant and another one, closer to the critical transition, in which U is temperature dependent. The details of the variation of U with T will be discussed in a separate paper. Here, we would like only to note that because of the temperature variation of U we are not allowed to use the Ngai model for relaxations in complex systems [21] to discuss the fits of fig. 1. Indeed in this model the value of the activation energy U (as well that of r0) has to be

A. Giannelli and C. Giovannella I Dynamics in high-T~ granular superconductors

oErl

i

i

-1 .41~I

-~

'

'

'

i i -.8

,

l

'

'

,

'

i , , , i , -.6 -.4 l

'

'

'

i

'

279

,

-.2 '

'

0

l

'

'

'

I__~

.2

o -1 ~.

"4~-I

~

.2

-.8

-.6

-.4

-.2

0

' ' ' i ' ' ' I ' ' ' I ' ' ' I ' ' ' ~

"

o~'I~,',,

-1

I

,

-.8

,

,

I

-.6 x'

,

,

,

I

-.4

(-1/4~)

,

,

-.2

0

Fig. 1. X' versus X" plots and relative fits for three different frequencies. The stars mark the break-down of the fit. The fit parameter (1 - n) values are: 0.275 (90 Hz), 0.285 (900 Hz), 0.315

(9000 Hz). considered a fictitious one, which accounts for the coupling between the single microscopic activated process and the low energy modes of the system and it would imply a constant U from which it would be possible to work out the true activation energy Us, U s = (1 - n)U. It is worthwhile noting that the values of U that we found in the region in which U is constant are included in the range, 20-100 meV, values that are very similar to those deduced for the activated motion of fluxons from other experimental techniques [22, 23]. It should be beared in mind that U is only an average value and more precisely the expression of the susceptibility should be written as: X* = f g(r, to) d~-/(1 + ito~-) = 1/(1 + itor) (1-~) . 0

Another hypothetical consequence of the observation of the skewed curves reported in fig. 1 is the occurrence of a stretched exponential relaxation in the time domain: qb(t) : qbo exp[--(t/7")(1-n')] ,

280

A. Giannelli and C. Giovannella / Dynamics in high-T c granular superconductors

such that f

X* = J exp(-itar) (-d~b/dt) d t , 0

with the expectation of ( 1 - n ) = ( 1 - n'). We remind the reader that in the present experimental conditions the stretched exponential law is due to the thermal fluctuations at the quasi-equilibrium and not to the gradient of the magnetic field, like in the high-field regime [24]. To check the "stretched exponential hypothesis" we have inverted the above integral and plotted the results for different values of (1 - n') in fig. 2. We can immediately note that the main features of the experimental data are all reproduced, i.e. with the decrease of (1 - n') the peak of the g" gets smeared out and shifts towards longer ~-(i.e. lower temperature, remember that the curves of fig. 2 are plotted as functions of the product tar). However, the quantitative comparison with fig. 1 is not completely satisfactory, because the value of ( 1 - n') needed to fit those curves is around 0.4. ~-~

.~

'~'

.6

~'~

.4

X

.2 0

I , , ,

0

~

10

20

30

40

.2 o

,

,

,

0

I

,

,

,

,

10

I

,

20

,

,

,

I

,

,

,

,~1

30

40

~OT

~

.2 0 0

.2

.4

.6

x' (a/4~r)

.8

1

Fig. 2. Plot of the computed X' and X" versus (o~-and X' versus X" as derived from the "stretched exponential hypothesis". In the X' versus X" plot the curve obtained for (1 - n ' ) = 0.4 is drawn in bold. With respect to the experimental data the X' scale is shifted by one in the positive x-direction.

A. GianneUi and C. GiovanneUa / Dynamics in high-T c granular superconductors

281

3. Critical dynamics In the previous section we have seen the effect of the frequency of the ac field on the non-Debye relaxation. Let now discuss its influence on the critical dynamical temperature. The stars in fig. 1 mark the departure of the experimental curve from the fit; it occurs at a temperature that corresponds to a rather abrupt change in the value of the activation energy and we think that it corresponds also to the dynamical breaking of the systems, i.e. the dynamical percolation of the cluster of non-correlated junctions. The critical temperature depends on the frequency and does not correspond exactly to the experimental peak of the X" [25]. After having obtained by extrapolation the critical temperature at zero frequency, we can test the reliability of the scaling hypothesis in zero magnetic field [26]: f = f0 [ T ( f ) - T(0)] z~, see fig. 3. From the linear fit we obtain z u = 6.8, which is a reasonable value for a "spin glass"-like transition described by the fractal cluster model [27, 28]. In fact, under the hypothesis of ref. [28] and knowing that ~b = 3.45 [9] we obtain 8 = ~b/(4 - 4)) = 6.27, and as a consequence z~, = 88/(8 + 1) = 6.9 is in very reasonable agreement with the above experimental determination. With v = 4/ 3 [9, 28] we obtain z = 5.2. Immediately we can derive also the theoretical value of the other critical exponents: fl = 4/(1 + 8 ) = 0 . 5 5 , 3' = 4 ( 8 - 1)/ (8 + 1) =2.89, ~ = 2 - 3'/v = -0.17, of the fractal dimension: D =3(/3 + 3')/ (2/3 + 3 ' ) = 2.58, and of the algebraic prefactor, A, of a possible power-law relaxation at the equilibrium below Tg (which remains to be verified): A = 1/ 28 = 0.08. 4. Conclusion In conclusion we have shown that the quasi-equilibrium dynamics of the intergranular junction network presents features very similar to those observed

A

3,5

bv

0

2,5

0,1

0;2

0;3

014

log (Tc(t)-Tc(0)) Fig. 3. Verification of the scaling hypothesis: plot of l o g ( T o ( f ) - To(0)) v e r s u s l o g f . To(0) = 88 K.

282

A. Giannelli and C. Giovannella / Dynamics in high-T,, granular superconductors

for a large v a r i e t y o f d i s o r d e r e d a n d glassy s y s t e m s , a n d in p a r t i c u l a r t h a t t h e f r e q u e n c y d e p e n d e n c e o f t h e d y n a m i c a l critical t e m p e r a t u r e c a n b e d e s c r i b e d by the scaling hyphotesis, with a value of the zu product c o m p a r a b l e with those f o u n d in spin glasses a n d p r e v i e w e d b y t h e f r a c t a l c l u s t e r m o d e l for a f r a c t a l d i m e n s i o n D = 2.58.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

[20] [21] [22] [23] [24] [25] [26] [27] [28]

R.H. Parmenter, Phys, Rev. 154 (1967) 353. T.D. Clark, Phys. Lett. A 27 (1968) 585. P. Pellan, G. Dousselin, H. Cortes and J. Rosenblatt, Solid State Commun. 11 (1972) 427. J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6 (1973) 1181. J.G. Bednorz and K.A. Muller, Z. Phys. B 64 (1986) 189. K.A. Muller, M. Takashige and J.G. Bednorz, Phys. Rev. Lett. 58 (1987) 1143. C. Giovannella, C. Chappert, P. Beauvillain and G. Collin, Int. J. Mod. Phys. B 1 (1987) 1011. C. Giovannella, P. Rouault, I.A. Campbell and G. Collin, Int. J. Mod. Phys. B 1 (1987) 1017. C. Giovannella, Phys. Stat. Sol. (b) 154 (1989) 273. C. Giovannella, in: Universalities in Condensed Matter Physics, R. Jullien, R. Peliti, R. Rammal and N. Boccara, eds. (Springer, Berlin, 1988), p. 93. J. Rosenblatt, C. Lebeau, P. Peyral and R. Raboutou, in: Josephson Effects - Achievements and Trends (ISI, Torino, 1985) p. 320. L.B. Ioffe and A.I. Larkin, Sov. Phys. JEPT 54 (1981) 378. D.R. Browman and D. Stroud, Phys. Rev. Lett. 52 (1984) 299. I. Morgenstern, K.A. Muller and J.G. Bednorz, Z. Phys. B 69 (1987) 33. A. Giannelli and C. Giovannella, in: Proc. SATT 3 Conf., Genova, Feb. 1990 (World Scientific, Singapore), in press. C. Giovannella, C. Lucchini, B. Lecuyer and L. Fruchter, IEEE Trans. Magn. 25 (1989) 3521, the ac susceptibility probe is described in the abstract contained in the abstract-booklet. L.P. Levy and A.T. Ogielski, Phys. Rev. Lett. 57 (1986) 268. C. Lucchini, C. Giovannella, R. Messi, B. Lecuyer, L. Fruchter and M. lannuzzi, Phys. Stat. Sol. (b), submitted. S.D. Murphy, K. Renouard, R. Crittenden and S.M. Bhagat, Solid State Commun. 69 (1989) 367; D.X. Chen and R.B. Goldfarb, J. Appl. Phys. 66 (1989) 2489; H. Dersch and G. Blatter, Phys. Rev. B 38 (1988) 11391; V. Calzona, M.R. Cimberle, C. Ferdeghini, M. Putti and A.S. Siri, Physica C 157 (1989) 425. F. Gomory, S. Takacs, P. Lobotka, K. Frohlich and V. Plechacek, Physica C 160 (1989) 1; P. Berg and G. Kozlowski, preprint; J.R. Clem, Physica C 162-164 (1989) 1137. See for example: Relaxation in Complex Systems and Related Topics, I.A. Campbell and C. Giovannella, eds. (Plenum, New York, 1990), in press. K.L. Ngai, Commun. Solid State Phys. 9 (1979) 127. R.H. Koch and A.P. Malozemoff, Proc. 1st Int. Symp on Superconductivity, 1988. Y. Yeshurun and A.P. Malozemoff, Phys. Rev. Lett. 60 (1988) 2202. C.W. Hagen and R. Griessen, Phys. Rev. Lctt. 62 (1989) 2857, and references therein. A. Giannelli and C. Giovannella, in: Relaxation in Complex Systems and Related Topics, I.A. Campbell and C. Giovannella, eds. (Plenum, New York, 1990), in press. Z. Koziol, Physica C 159 (1989) 281, and references therein. M. Continentino, Phys. Rev. B 34 (1986) 471. I.A. Campbell, Phys. Rev. B 37 (1988) 9800.