Physica C 156 (1988) 807-816 North-Holland, Amsterdam

EXCESS ELECTRICAL CONDUCTIVITY IN POLYCRYSTAIJJNE Bi-Ca-Sr-Cu--O C O M P O U N D S AND T H E R M O D Y N A M I C FLUCTUATIONS OF T H E A M P L I T U D E OF T H E SUPERCONDUCTING ORDER PARAMETER

F61ix VIDAL, J.A. VEIRA, J. MAZA and J.J. PONTE Laboratorio de Fisica de Materiales, Unlversidad de Santiago de Compostela, 15700, Spain

F. GARC|A-ALVARADO and E. MORAN Departamento de Qulmica Inorg,~nica, Facultad de Quimica, Universidad Complutense, 28040 Madrid. Spain

J. AMADOR, C. CASCALES, A. CASTRO, M.T. CASAIS and I. RASINES lnstituto de Ciencia de Materiales, CSIC, Serrano 113, 28006 Madrid, Spain

Received 6 October 1988 Revised manuscript received 28 October 1988

Measurements of the rounding effects on the electrical resistivity above the superconducting transition in Bi-Ca-Sr-Cu-O polycrystalline compounds are reported, to our knowledge for the first time in this HTSC system. These effects are analyzed in terms of thermodynamic fluctuations of the amplitude of the superconducting order parameter (SCOPF). In the mean-field-like region, the experimental critical exponent seems to be compatible with an order parameter of two components (2d) fluctuating in two dimensions (2D). This contrasts with previous results for A-Ba-Cu-O (A--Y, Ln) and Ln-M-Cu--O (M=Ba, St) su- perconductors, where SCOPF seem to be 2d-3D in all the different dynamic critical regions.

1. Introduction

In this work, we present high-resolution measure- ments of the rounding effects on the electrical resis- tivity, p (T ) , around the normal-supereonducting transition in various polycrystalline B i - C a - S r - C u - O samples (BSCO system). Then, the excess- or para- conductivity above Tc, Aa(T) , will be extracted from these data and analyzed in terms of the thermody- namic fluctuations of the amplitude of the super- conducting order parameter (SCOPF).

Although the rounding effects on p ( T ) have al- ready been studied in the A - B a - C u - O (ABCO, A = Y or rare earth) HTSC system [ 1], and some preliminary results have also been obtained in the L a - B a - C u - O (LBCO) HTSC family [2], to our knowledge it is the first t ime that these effects are measured in the BSCO system. As is well known, the differences among these three HTSC systems con- cern such basic aspects as their crystallographic

structure or their general electronic properties, in- cluding the dramatic differences among the values of their corresponding normal-superconducting tran- sition temperatures. To probe the influence of these differences on the rounding effects on p ( T ) around Tc, and probably indirectly on the nature of the cor- responding order parameter is, therefore, of great interest.

Let us already stress in this Introduction that the presence of non-intrinsic rounding effects (associ- ated with, for instance, sample inhomogeneities and polycrystallinity) in p ( T ) introduces some ambi- guities and uncertainties on both Aa and Tc, the ("physical") bulk critical temperature (see below). The separation of the nonintrinsic rounding effects (always present in real measurements) from the in- trinsic critical effects associated with SCOPF is a very general and difficult problem not only when analyz- ing the critical behaviour near Tc in superconduc- tors but in critical phenomena in general [3,4].

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808 F. Vidal et al. / Excess electrical conductivity in B i - C a - S r - C u - O compounds

However, we will see here that in spite of these dif- ficulties our measurements will probably allow us to observe the main intrinsic aspects of At7 in the BSCO system.

In comparing our data with the theoretical ap- proaches for SCOPF, we will concentrate our anal- ysis on the mean-field-like range, where the general theoretical results based on the time dependent Ginzburg-Landau (TDGL)-like approaches are very probably applicable [5]. In addition, the Aslama- zov-Larkin (AL) theory provides an approach of SCOPF based on the mean field BCS theory [5,6]. Also, the Lawrence-Doniach (LD) theory for 2D layers coupled by the Josephson effect may, in prin- ciple, be used in the mcan-fidd region [ 5,7 ]. We must already note here, however, that probably only the general and qualitative aspects of these approaches may be used in HTSC [ 8,9 ]. In addition, due in par- ticular to uncertainties on the precise values of the different magnitudes arising in these theories, the analysis of HTSC in the mean-field-like region may present some ambiguities [ 10]. But, in any case, it is in this region of "moderate" values of the order parameter correlation length, ~(T), where the influ- ence on A~ of other characteristic distances of each HTSC system (like the distance between the Cu-O planes) are expected to be relevant. In fact, probably the more interesting and new result of the present paper is to show, to our knowledge for the first time, the important differences for Ao(T) in the three main HTSC systems discovered until now. We will see here that when analyzed in terms of SCOPF, these dif- ferences seem to indicate that in the mean-field-like region SCOPF in the BSCO system are 2D, whereas they are 3D in the LBCO and in the ABCO systems. Although we cannot exclude some ambiguities in our analysis, in part associated with the presence of spu- rious rounding effects on p (T) , we believe that these basic differences are real.

2. Experimental details

Two series of polycrystalline samples were cut from two different BSCO pellets (here noted A and B) of nominal composition Bi2Ca2Sr~Cu.,O,. (pellet A) and Bi2Ca~Sr2Cu2Or (B), with y~, 8. The preparation procedure followed in both cases the main lines of

the process used in previous works [ 1 l - 13 ]. Let us, however, present here the main aspects: Pellet A was prepared from a stoichiometric mixture of analytical grade Bi203, CaCO3, SrCO3 and CuO, which was heated in air for 12 hours at 850°C, ground, pelle- tized, reannealed for 12 h at 860°C and slowly cooled down to room temperature. From X-ray diffraction patterns and scanning electron microscopy, the sam- ple, which was black, appeared as single-phase with grain size d s < l ~tm. All the reflections were indexed in an onhorombic unit cell whose parameters are a=5 .393(4) A, b=27.12(2) A, c=30.72(4) A and I / - 4 4 9 6 (6 ) A 3. Other details of samples A will be published elsewhere.

The preparation and some of the basic parameters of samples B were already described elsewhere [ 14 ]. Let us, however, note here that these samples were prepared from stoichiometric mixtures of CuO, Bi203, CaCO3 and strontium acetate which were heated in air at 830°C during two days, reground, pressed into pellets, sintered again at the same tem- perature and, finally, rapidly quenched to room tem- perature. Powder X-ray diffraction pattern again corresponds to a orthorhombic phase similar to that described in previous works [ 12-16] for Bi2Sr3_xCa~Cu2Os+~. The resulting parameters are a=5.356 A, b=5×5 .393 A and c=30.714 A. Scan- ning electron micrographs of the fractured sample are shown in fig. 2a of the paper by Michel et al. [ 14 ]. They are composed of platelets, with lamellar mi- caceus aspects and they have an average composi- tion for the superconducting phase of Bi2.5oSrL44CaHoCu20.,.. Others aspects of the stoi- chiometric, structural or electronic structure of these B samples are described in the paper by Michel et al. [14].

Electrical resistivity (p) measurements were made with a four-probe method also described earlier [ 17 ]. Most of the present measurements were performed by using current densities from 0.5 to 1 A/cm 2 (less than 30 mA through the various samples) and our experimental system was able to detect changes o fp of the order of I ~ cm. As in previous experiments in the ABCO system [ 1,10 ], special care was taken to control the sample temperature stability and ho- mogeneity, and relative temperature variations were resolved to better than 10 mK by using Pt-100 and Rh-Fe sensors. The "ideal" reduced-temperature

F. Vidal et al. / F.xcess electrical conductivity in B i -Ca-Sr -Cu-O compounds 809

resolution will be, therefore, of the order of 10 -2 K / Tc (K) . However, the presence of non-intrinsic rounding effects (associated with, for instance, stoi- chiometric inhomogeneities and polycrystallinity) decrease such a reduced-temperature resolution: First, these effects broaden the bulk Tc, which will be an average temperature over all of the sample. Then, these effects, which also deform p ( T ) in the vicinity of To increase the uncertainties associated with our extraction of the "physical" bulk Tc from these curves (see below). The lower limit for the re- duced-temperature resolution will be of the order of A T / T o where ATis the half-width of the p ( T ) curves [ l ]. A s / I T is typically of the order of 4 K for the samples studied here, this lower limit will be of the order of 5 × 10- 2. We must stress here that even with such a very poor reduced-temperature resolution we should be able to observe most of the mean-field-like region for Atr. But, however, the practical resolution will be between the "ideal" and the lower limit. We estimate that this is around the dispersion among the distinct Tc that may be defined (see below). This dispersion is of the order of I K, so the practical re- duced-temperature resolution is estimated to be of the order of 10-2.

3. Experimental results and data analysis

3.1. General aspects

Typical examples of p ( T ) are shown in figs. l and 2a and 2b (in fig. 1, p (T) was normalized to p at 300 K). Triangles and circles correspond to, respec- tively, samples A and B. To compare with previous results for ABCO HTSC, the squares curve in fig. 1 corresponds to a typical YBCO polycrystalline sam- ple [ l, 10 ]. The excess electrical conductivity, Atr, is extracted from this type of curves by applying the standard definition [ l ]:

1 1 A~( { ) -p (e~ PB(¢) ' ( l )

where p(~ ) and PB ( e ) are respectively the measured and the background electrical resistivity, and

T - Tc e - Tc ' (2)

a o

\

r-

1

8 . 8

8 .G

0 . 4

0 . 2

0

o A o ~

o

,_~ ~o. _~, 5 0 100

O 8i2 St2 C% Eu20e.~

r~ YIBa2 CuaOT-b

t i i i I

1 5 0 200 250

T ( K )

} 0 0

Fig. 1. Typical examples of the resistivity (p) versus temperature (T) curve for two BSCO samples (A triangles, B circles), with j= 0.4 A/cm 2. For comparison also shown is a typical p(T) curve ofa YBCO sample (squares), with j r 0. l A/cm 2. The solid lines are the noncritical background resistivity (Pa) obtained as indi- cated in the text. To allow a clearer observation ofps(T), only about 5% of the data points for each sample have been plotted.

is the reduced temperature, and Tc is a normal-su- perconducting transition (critical) temperature which must be defined in each case. This paper is centered on the case where Tc will be the mean-field critical temperature.

The extraction of Ao(e) from the p ( T ) curves is probably the central and more difficult part of the data analysis. Here we will follow the same proce- dure that we have already used to extract A~r(~) for the ABCO HTSC [l ,10] and which may be sum- marized as follows. First, we estimate the back- ground or noncritical part of p(T) by taking into account that, for all the BSCO samples studied here, p ( T ) follows a linear temperature dependence from at least 2Tc to room temperature, as in the examples of fig. 1. This metallic behavior in the normal state which appears clearly illustrated in fig. l is similar to that observed, as is well known, in polycrystalline ABCO samples. Thus, PB (T) in this work will be the resistivity linearly extrapolated from the p ( T ) data above 2Tc (the straight line in figs. l and 2a and 2b). We have checked, in addition, that eventual small errors (less than 5%) in Pa(e) and p ( T c ) affect Atr appreciably only very close to, or very far away from, Tc (see below).

The second main point for the extraction of the experimental Aa concerns the choice of its func-

8 1 0 F. Vidal et al. I Excess electrical conductivity in B i - C a - S r - C u - O compounds

O

E v

2 o_

50

E 4 O

E ~J

o_ 2

O 5B

i i i i i I

IIi2 St2 CaiCuz Oi.b ~ ~ ~.

- - Tc~ """ k TCM . Mlon-field-likl

. . o O "" . . . . . . . . : : : 5 ~ . , : n , , , , "'"'l ...... ; " ' " ] ...... ; .... 68 70 88 90 100 118 2B

T ( K )

i = I i i t

el2 S 5 Col Cu2 (]B*b o o . q . . ° ° ° ~ ~ ° 5

8 g

g g

(b) o o

o o : ~ = = ~ - . ~ n I I I i I i I i

60 7~ 80 90 108 118 20

T ( K )

G O Y] 802 Cu 3 07.

. . . . . . . . . - - - - 7 - -

~-^L " l f l I /: Ig-I ,.~-~,.l~l,,. I E r e s i s t i v i t y , ~ . . / : [ " [ i x

Q_ 2o T~ <mJ i i i ~ ~-̂ L i - ~ . . . . . t i v i t y ( C )

) .::: ....

85 9~ 95 00

T ( K )

Fig. 2. Typ ica l e x a m p l e s o f p(T) a r o u n d the s u p e r c o n d u c t i n g

transition. The dotted line is dp/dTin arbitrary units. The dashed line is the background resistivity. The different transition tem- peratures are obtained as indicated in the text. (a) This example corresponds to a BSCO sample (same data as the circles in fig. 1 ). The solid line is p(T) obtained by subtracting from PB(T) the Aslamazov-Larkin (AL) excess conductivity for fluctua- tions in two dimensions (2D). (b) The same experimental data as in (a) compared now with the AL theory for fluctuations in three dimensions (3D). (c) This example corresponds to a YBCO sample (the same data as those represented by the squares in fig. l ). The solid and dotted--dashed curves were obtained from the AL theory for respectively 3D and 2D cases. For details see the text.

tional dependence on e, because this will allow the estimate o f the corresponding critical temperature, Tc. In the absence o f any specific theory for critical phenomena in any HTSC system [8,9], we will ap- proximate Aa(e ) by

Aa - - = A ( x , ( 3 )

~(Tc)

where A is a temperature independent amplitude, x is a critical exponent and the normalization factor, t r(Tc), is the conductivity at To A and x will depend on the critical ~-region studied and, indeed, also on the precise choice o f Tc in e. As is well known, sin- gularities near critical transitions in real physical sys- tems cannot always be adequately described by pure power laws [ 18 ]. In addition, Aa is very probably influenced by various non-intrinsic effects, as noted before [ l, 10 ]. However, the functional dependence o f eq. (3) is that predicted in the mean-field region by both the Aslamazov-Larkin (AL) theory for SCOPF based on the microscopic BCS approach, and also by the approach based on the t ime-dependent Ginzburg-Landau models (see, for instance, ref. [5 ] ). Also very general dynamic scaling ideas sug- gest that eq. (3) can be a reasonably good approx- imation for Aa in the so-called "crossover" and "full critical" regions close to Tc [18-20] .

Before presenting the excess conductivity let us comment here on two other aspects o f its extraction. The first one concerns the determination o f the bulk (average) To As in previous analyses of other HTSC [ l ,10 ] , a first indicative (non-physical) critical temperature is the so-called resistive critical tem- perature TcR, defined by [ l ]

p ( T c R ) = ½ P a ( T c R ) , (4)

i.e., TcR is the temperature for which the measured resistivity is half the background resistivity. Another indicative critical temperature is To, the tempera- ture at which p ( T ) around the transition has its in- flexion point. Although both temperatures do not have any physical significance, it seems reasonable that they must be close to each other and to any in- trinsic physical temperature as well, the central one in this paper being the "mean-field transition tem- perature", TcM, associated with the Aslamazov-Lar- kin and with the T D G L mean-field theories (see below). The locations o f these three temperatures on

F. Vidal et al. / EJccess electrical conductivity in Bi-Ca-Sr-Cu-O compounds 811

p (T) are indicated in fig. 2a for one of the examples shown in fig. 1. Their relative closeness is an indi- cation of the consistency of the procedure used to their extraction from the experimental data (includ- ing the use of linear temperature dependence for Pa), as well as an indication of the absence of an appre- ciable influence (to within our resolution) of the tail of p (T ) below Tc on our estimate of the different Tc (however, such a tail would slightly modify the ex- cess conductivity amplitude, see below).

The other general aspect to be commented on here concerns the influence of the presence of spurious (nonintrinsic) effects on the observed rounding of p (T) above Tc, for example those associated with the polycrystallinity of our samples. As noted in the Introduction, the separation of non-intrinsic round- ing effects (always present in real measurements) from the intrinsic critical effects is a very general and difficult problem not only when analyzing the crit- ical behavior near Tc in superconductors but in crit- ical phenomena in general [ 3,4]. However, we must stress here that, in any case, the non-intrinsic effects arise in the critical behavior after the intrinsic ones [4]. So, the possible rounding of p (T ) associated with polycrystallinity or inhomogeneities will be probably strongly mitigated by the "previous" pres- ence of intrinsic SCOPF rounding effects in the in- dividual crystallites. We have also checked, finally, that the possible presence of a tail of p (T ) below Tc (associated for instance with weak coupling between grains) as important as 50% ofpa(Tin) does not af- fect either our extraction of TCR or Tc~ or that of the excess conductivity critical exponent (to within the above indicated uncertainties). In contrast, such a tail slightly modified both the background resistivity and the excess conductivity amplitude (see below).

3.2. Quafitative analysis

A first qualitative, but very illustrative, check of the adequacy of the analysis of the rounding effects on p ( T ) above Tc in terms of conventional SCOPF effects, and also of the influence of the possible non- intrinsic rounding effects, may be easily obtained di- rectly by comparing p (T ) extracted from eqs. ( 1 ) - (3) with the measured one. From eq. ( 1 ) we get

p (T ) =pB(T) [1 +pa(T) Art(T) ] - ' , (5)

where Ps(T) is the background obtained as indi- cated above and Aa(T) is given by eq. (3). Some examples of the comparison of eq. (5) with the mea- sured p (T ) are presented in figs. 2a-2c. The solid line in fig. 2a was obtained by fitting eq. (5) to the experimental data with A and x as free parameters but using Tc~ in e. The limits of the fitting region above To were obtained by imposing the fitting to be of reasonable quality. The quantitative criterion was a r m s deviation of the order of I% or better, which typically corresponds to a variation o f x in eq. (3) (the "critical" exponent, which is the relevant parameter here) of less than 0.05 (to be compared withx= - 0.5 in 3D or x = - I in 2D predicted by the TDGL approaches). The resulting values of the free parameters in the indicated region, noted "mean- field-like region" in fig. 2a, are A = 7 . 2 × l0 -2 and x = -0 .95 , with a rms deviation of 0.3%. This value o f x is indeed very close to that expected for SCOPF in the mean-field region of an order parameter of two components (2d) fluctuating in two dimensions (2D).

Before analyzing quantitatively the above basic re- sult, let us check if the qualitative procedure that we are using until now is capable of discriminating be- tween x = - ½ (3D) and x = - l (2D). One of the qualitative tests we have followed for that purpose is summarized in figs. 2b and 2c. The two lines in figs. 2b were obtained by fitting eq. (5) to the ex- perimental data in the same region as in fig. 2a, with A as a free parameter but with x = - ½ and using To in e (solid line), and with A and Tc as free param- eters but with x = - ½ (dotted-dashed line). We see that whereas the fit to the experimental data in fig. 2a agrees very well with x = - 1 (2D), the two curves with x = - ½ in fig. 2b do not agree with these ex- perimental data.

Since a qualitative indication of that seems to im- ply fundamental differences between the ABCO and the BSCO critical dynamics in their corresponding mean-field region, in fig. 2c we show an example of Aa in a YBa2Cu3OT_~ polycrystalline sample already studied in our laboratory [ 1 ]. The solid line in this figure was obtained from eq. (5) by using the cor- responding To in ~ and with A and x in eq. (3) as free parameters. The agreement in the mean-field re- gion is excellent, the rms deviation being ~0.7%. Note that the reduced temperature limits of this re-

812 F. 1,'idal et aL / Excess electrical conductivity in Bi-Ca-Sr-Cu-O compounds

gion agree well with those expected from the Ginz- burg criteria in 3D [ 20 ]. The corresponding values for the free parameters are A = 9 . 6 X I 0 -3 and x = - 0 . 5 2 (the latter to be compared with x = - ½ for 3D). The dotted-dashed line was obtained from eq. (5) but imposing x = - 1 with A and Tc as free parameters. The disagreement between the experi- ment and theory in the expected mean-field region is evident.

The main conclusions of the qualitative compar- ison summarized in figs. 2a-2c are (i) The observed rounding of p ( T ) above Tc may be explained, at least at a qualitative level and not too close or not too far from Tc (in the mean-field-like region), by eqs. (3) to (5) by using Tct as critical temperature and a re- sistivity background obtained as indicated before (in particular, linear in T). (ii) For the B i - S r - C a - C u - O system studied in this paper, Aa in the mean-field region is well accounted for by using x = - 1, i.e. in terms of 2D-SCOPF effects in the dirty limit (i.e., only taking into account the AL contribution). This contrasts with that found for the Y - B a - C u - O sam- ple (and for all ABCO bulk HTSC samples studied until now, with A = Y or rare earths), where x ~ - in the mean-field region and where, therefore, SCOPF seem to be 2d-3D. (iii) The observation of these systematic differences for both HTSC systems is a nice indication that this analysis procedure, al- though qualitative, is able to discriminate between different critical exponents. It is also an indication that these exponents are intrinsic to these HTSC and that they seem not to be associated with non-intrin- sic properties (polycrystallinity, macroscopic in- homogeneities). Therefore, these macroscopic (over many interatomic distances) non-intrinsic effects, which in any case would arise after those of the SCOPF [4 ], seem not to play a relevant role on p (T) above Tc.

3. 3. Quanti tat ive analysis

Like for the other HTSC systems [ 1,2], a quan- titative comparison of the measured Aa in the mean- field-like region with T D G L or with the AL or LD- like approaches [ 5-9 ] is facilitated by plotting In Aa as a function of In ¢. In fig. 3 we show two examples of such a representation, circles corresponding to the data of figs. 1 and 2a and 2b (a sample of type B)

t-T c -3 -2 -i

1~ T c 1 0 T c 1~ T c T c

~_u -2

oo ~Ooo o~"~.-

- - 6 - 4 - 2 O.

Ln [ ( T - T c , ) / T c , ]

1 ..... 2 I I I I 1 5 0 5 1 8 5 2 1

Fig. 3. Log-log plot of the normalized (to ~ at Tc, ) excess con- ductivity versus reduced temperature, the latter obtained by us- ing Tc~. Circles and triangles correspond to two BSCO samples, whereas diamonds and squares correspond to two YBCO sam- ples. The ~/¢(0 ) abscissa assumes the conventional dependence for ~(~), i.e., ~(~ ) = ~ ( 0 ) ~ - t/2 and that all the anisotropy of these compounds is contained in ~(0).

and triangles corresponding to a sample of type A, with ~ obtained by using their corresponding Tel as critical temperature. The other two curves of this fig- ure correspond to two YBa2Cu307_6 samples [ l ] and are presented here for comparison. In all these curves, Aa(e ) is normalized to dr( Tel ), in contrast with pre- vious works in other HTSC systems where as ( T = 300 K) was currently used as the normalization factor. In addition to the fact that a(T¢~) is a well defined characteristic conductivity arising in the studied temperature region, we have checked now that its use reduces the differences of the excess conductivity amplitude (A in eq. 3) of the different compounds of each HTSC family.

The results of fig. 3 are of central interest. They show, first, that Aa(~) for two different polycrystal- line samples are very similar, even for In ~ < - l, where the influence of local defects are expected to be more important as a consequence of the short co- herence length in this "high" reduced-temperature region. The kink of Aa around In e = - l, clearly shown in fig. 3, and also in fig. 4 (see below), cor- responds to a very small drop in the resistivity around 1 l0 K, which for the compounds studied here is hardly perceptible in fig. 1. Such a resistivity drop

F. Vidal et al. / Excess electrical cOnduclivi(v ~It Bi~Utlt~t- CU-O coml~nds 813

-l

l~ Tc Tc l

C - 4 J

- 3 - 2 -I 0

L n [ ( T - Tc.)/TcM ]

I I I 5 2 1

Fig. 4. An example of a log-log plot of the excess conductivity versus reduced temperature, this last obtained by using TcM, the 2D mean-field-like critical temperature. For details see the text.

around 110-120 K has been often observed in this HTSC family (see e.g. Mackay et al. [21 ] ) and is accompanied by an anomaly in the magnetic data [ 21 ], so it is attributed to the presence of a small fraction of a higher Tc phase.

As it was the case for the ABCO-HTSC system studied previously [ 1,2 ], the general aspects of the curves presented in fig. 3 are found for all the BSCO samples (A and B) and current densities used. This seems to suggest that Air above Tc in polycrystalline BSCO HTSC is also driven by SCOPF in individual grains and anyhow independent of intergrain links. These effects seem to be, therefore, intrinsic to BSCO superconductors.

A second interesting result of fig. 3 are the simi- litudes and differences for Atr in BSCO and ABCO HTSC. In this figure, squares and diamonds corre- spond to two different polycrystalline YBCO sam- ples previously studied by us [ l ]. In what concerns the similitudes, we may identify in both systems dis- tinct regions for Aa. Detailed analysis of these re- gions in terms of the GL theory.and of the dynamic scaling approaches for a 2d order parameter has been presented elsewhere for ABCO system [l ]. For brevity, let us just note that in both cases the mean- field regions correspond (qualitatively) to those covered by the straight lines. On the right side of this region it appears, in both systems, a "high temper-

ature" region where Art(e) does not follow, even qualitatively, eq. (3). This "high temperature" re- gion is very probably associated with the low values of the different components of the correlation length, and it will be analyzed elsewhere. On the left of the mean-field region, Aa in both systems still follows eq. (3), but with different critical exponents. In the ABCO system these regions closer to Tc are associ- ated with the crossover and critical regions (in 3D), for which x = - ] and x = - ], respectively. In the BSCO system these variations of the critical regime will probably be also present, but they will be masked by a possible crossover from 2D to 3D behavior, as a consequence of the increase of the ratio of the cor- relation length to the Cu-O interplain distance. Such a dimensional crossover has already been observed in YBCO thin films [22]. However, due to the ex- perimental uncertainties in these regions very close to Tc, and due also the absence of theoretical de- velopments in HTSC, we will not comment further on such a result here.

The differences for Aa in both HTSC systems are also clearly displayed in fig. 3. Since there are quan- titative theoretical results for Atr in the mean-field region on the grounds of the GL theory, probably ap- plicable to HTSC [5-9] , we will focus our analysis here on that region. The strength of Art in this region and also its reduced-temperature location are very different for both families of HTSC. But its reduced- temperature dependence is clearly very different too, as expected from the qualitative analysis presented above, in terms of eq. (3): the critical exponent for the YBCO sample is of the order of - ½, whereas for the BSCO samples the critical exponent in that re- gion is very close to - I. As noted also in the above subsection, on the grounds of the conventional TDGL-like approaches a critical exponent x = - l indicates SCOPF in two dimensions (2D), whereas x = - ½ indicates SCOPF in 3D, as observed for the ABCO system. This is indeed a very important and, to our knowledge, new result. So, in order to check the possible influence of Tel used in fig. 3, we obtain the "mean-field critical temperature", ToM, by using a very simple but precise procedure, already used by us to analyze the ABCO system [ 1 ]: we first plot tr( To ) /Aa as a function of T. In this way, we obtain graphically the limits of the mean-field region and the corresponding values of A and TcM. Next, to ob-

814 F. Vidal et al. / Excess electrical conductivity in B i -Ca-Sr -Cu-O compounds

tain the final values of A and TcM, Aa/a( Tc~ ) inside that region is fitted to eq. (3) with A and TcM as free parameters, but with the constraint x = - 1. The value of TcM for the example of fig. 2a is indicated in that figure. As expected, TcM is very close to both TcR and Tel.

A typical example of the resulting In Aa vs In ~ in the mean-field region, with ~ = ( T - TcM ) /TcM, for a BSCO sample (B) is shown in fig. 4. The solid line corresponds to x = - 1 and A = 7 . 3 × 10 -2. This am- plitude may be compared with that predicted by the Aslamazov-Larkin (AL) or with the Lawrence- Doniach (LD) theories in 2D. For the 2D-AI theory,

e 2 p(T¢l) A ( A L ) - 16h d ' (6)

where d is a characteristic length of the two-dimen- sional system. By using the above indicated value of A, we obtain d - ~ 500 A, which is indeed much larger than either the typical distance between the Cu-O planes ( -,- 12 A) or the expected values of ~(~) in that region if we suppose ~o± - 7 - 8 A, like proposed for ABCO compounds [23]. Important disagree- ments between the observed A and that obtained from the AL theory have been already observed for the 3D SCOPF in the ABCO system (also in the mean-field region) for samples having low electrical conductiv- ity in the normal state [ 1 ].

Although the influence of the important intrinsic anisotropy of these materials cannot be excluded, these differences for A seem not to be explained by the correction introduced by the conventional Law- rence-Doniach model for layered superconductors [ 7 ]. The LD amplitude in eq. ( 3 ) may be written in the case of 2D-SCOPF (i.e., with x = - 1 in eq. 3) a s

A ( L D ) = A ( A L ) [ 1 + roe-J ] -I/~-, (7)

where

PO ~

~ is the coherence length perpendicular to the Cu- O planes and d is the layer spacing. The important point here is that, as one can see immediately, in or- der to keep x,~ - 1 in eq. (3) (i.e., the 2D nature of SCOPF),

Po E - l << l ,

so the amplitude correction introduced by the LD model cannot account for the observed disagreement with the experimental amplitude of Aa in the mean- field-like region. In fact, as we have observed in the other HTSC systems studied until now [ 1,2], A seems to be much more affected than x by local ef- fects, stoichiometric inhomogeneities, local varia- tions of the electrical current density, pair breaking effects, etc. Moreover, the amplitude of Aa may de- pend on the atomic disorder (localization effects [24] ). We shall analyze these effects on A elsewhere. However, concerning the probable presence in ce- ramic samples of nonuniformities of the electrical current density, we must note here that such an ef- fect could make A (as defined by eq. 3) independent of a(Tc~), i.e. Aa proportional to the normal con- ductivity above Tel. In contrast, such an effect would very slightly affect the critical exponents, in agree- ment with what is experimentally observed in all HTSC systems studied in our laboratory. Let us stress here that the difficulties arising in analyzing the am- plitude concern all the critical phenomena in general [ 18,19 ]. So, unlike it has been done in other exper- iments [ 25 ], we will not use here our results on the amplitude of the critical phenomena to try to con- clude about the nature of the order parameter.

Unlike the ambiguities that remain forA, the crit- ical exponent for Aa in the mean-field region in BSCO samples seems to be well defined and equals - 1 . Such a behavior is compatible with a superconduct- ing order parameter of two components (2d) fluc- tuating in two dimensions (2D), i.e., 2d-SCOPF-2D. This contrasts with Aa in ABCO superconductors, which seems to be compatible with 2d-SCOPF-3D. These basic differences for Aa between both HTSC systems seem to be real, and they are perhaps as- sociated with the enhanced two-dimensional char- acter of BSCO superconductors: in the BSCO compounds, the distance between the Cu-O planes (those that seem to play the central role in the su- perconductivity of all known HTSC [ 26 ] ) is larger than in LSCO or ABCO superconductors [ 12-16 ]. In fact, the structure of the subcell in BSCO com- pounds is, as indicated by Sunshine et al. [ 15 ], con- siderably different from ABCO or LSCO materials, with infinite Cu-O planes widely separated ( ~ 12

F. Vidal et al. /Excess electrical conductivity in B!-(a-Sr-Cu-O compounds 815

A) by Bi -O double layers. This two-dimensional character o f the BSCO compounds seems also to be revealed by electronic band structure calculations [27,28]. In addition, in real compounds these planes seem to be much better defined in BSCO than in the other systems, as indicated by the very high anisot- ropy of the resistivity tensor in these materials [29] . In ABCO, the oxygen intercalation is probably in- homogeneous which, together with the larger ortho- rhombicity, may lead to microscopic twinning [29] . Finally, it is worth noting here that (as shown in fig. 3) in the mean-field region A u / o ( T o ) has similar amplitude for both types o f HTSC. In contrast, both corresponding regions are shifted in reduced tem- perature and ~ ( 2 D ) / ¢ ( 3 D ) - 1 0 . This qualitative feature for the relative behavior o f A o / a ( T a ) in both systems is that expected on the grounds o f the 3D and 2D-AL approach [20] , in spite o f the important disagreement noted before when comparing both the AL and the LD approaches with the absolute values for A. Infact, taking into account the above reduced temperatures for the 2D and 3D samples in fig. 3 having similar normal resistivities, we obtain the quotient d/~(O) ~ 1 which is not far from what should be expected [ 12-16,23] .

4. Conclusions

We have presented here the, to our knowledge, first measurements o f the rounding effects on the elec- trical resistivity in the high-Tc B i - C a - S r - C u - O sys- tem. The observed effects above Tc seem to be intrinsic to this type o f HTSC, even though the pres- ence o f nonintrinsic rounding effects, associated for instance with small stoichiometric inhomogeneities or with the polycrystallinity o f the samples, cannot be ruled out. These effects above Tc are attributed to the presence o f important thermodynamic fluc- tuations o f the ampli tude o f the order parameter (SCOPF) , as expected from the t ime-dependent Ginzburg-Landau-like approaches. In the mean-field- like region, the critical exponent seems to be com- patible with an order parameter o f two components (2d) fluctuating in two dimensions (2D) . This con- trast with previous results in ABCO, and also with our preliminary results on the 40 K class La 2_ ~rxCuO 4 (LSCO) superconductors [ 2 ], where

SCOPF seems to be, in these two classes o f HTSC, 2d-3D. However, the amplitude o f Aa in that region is much less than expected from either the AL or the LD theories for, respectively, isotropic or layered ideal BCS superconductors. Measurements o f Aa in single-crystals o f this HTSC system must, indeed, help to answer or to confirm some o f the points stud- ied in this work. However, we believe that the main general aspects o f Aa in the mean-field region in BSCO-HTSC are those presented here.

Acknowledgement

This work was not explicitly supported by the CAICYT (Spain) , but we have used financial sup- port f rom the project PR-84-620.

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