Measurements of the paraconductivity in the a-direction of untwinned Y1Ba2Cu3O7−δ single crystals

PhysicaC218 (1993) 257-271 North-Holland

Measurements of the paraconductivity in the a-direction of untwinned Y 1Ba2Cu307_6 single crystals A. Pomar ‘, A. Diax a,b, M.V. Ramallo 8, C. To&n *, J.A. Veira a and F&ix Vidal**c L LAFIMAS, Departamentode Fisica de la Materia Condens& Facultad de Fisica, Vniversidadde Santiago, Santiagode Compostela,15706, Spain b Lboratoire de Chimie des Solides. VA CNRS no. 446, Vniversitkde Paris&&B& 414. OrsayCedex, 91405 France c Laboratoirede Physiquedu Solide, VPR 5 CNRS, ikole Supkrieurede Physiqueet de Chimie Industrielles de la Villede Paris, 10 Rue Vauquelin.Paris Cedex 05, 75231, France Received 12 July 1993

We report detailed experimental results of the paraconductivity in the a-direction, Au,,,of two highquality YiBasC~sOr_~ single crystals having a w ( - 0.3 mm2) untwinned mgion in their centers, and with transition temperatures of 90.0 K and 90.8 K. The exceptional sharpness of their resistive transitions (their upper half widths are less than 0.15 K) allows us to obtain quantitative information of Au,,(e), which is not atkcted by the CuO chains, up to reduced temperatures, c, of the order of lo-‘. The general scenario compatible with our experiments consists of the assumption of ( 1): A mean field-like region (MPR) up to t = 2 x 10p2, where Au.,(e) may be explained in terms of the Lawrence-Doniach-like (LD) approach by taking into account the existence of two Josephson-coupled CuOr superconducting planes per unit cell. The characteristic superconducting length amplitudesinthisMFRareQ(O)=(1.1fO.2)~,and~(0)=(0.12~0.02)nm.(2)Inalgeementwithourearlierqualitativereeults, below t%2 x 10S2 the measured p.(T) separates from the mean-field behavior. Although so close to the transition we cannot exclude the influence of small stoichiometric inhomogeneities, these Au,(t) data for e S 2x lob2 may be explained, also on a quantitative level, on the grounds of the 3D-XY model, with a critical exponent of x= - f for AUK, and x= - 3 for fluctuationinduced diamagnetism. This scenario corresponds to one complex component order parameter, i.e., conventional is,+vave pairing or one complex component unconventional pairing.

1. Intro4lmtlon

High-precision data of the influence of the superconducting order parameter fluctuations (OPFs) on the electrical conductivity, a(T), and on the magnetic susceptibility, x( T), in the weak magnetic field limit, will help to understand the superconducting transition in copper oxide materials (HTSC ) [ l-4 1. In particular, the comparison of paraconductivity, Au, and of fluctuation-induced diamagnetism, Ax, may serve to check some general aspects of the superconducting pairing state and to obtain useful relationships between some of the characteristic lengths arising in the Ginzburg-Landau-like descriptions of the superconducting transition [ 51. Recently, we have presented detailed results of A& (for the ap; plied magnetic field, H, perpendicular to the ab plane) and of Ax, (for H parallel to the ab plane), in the weak magnetic field limit, obtained in two high-

quality Y,Ba2CuS0,_a single crystals [a]. The first aim of the present paper is to complete this information about Ax by presenting the paraconductivity along the u-axis (the direction in the CuOz planes that is not affected by the presence of CuO chains along the b direction), of two single crystals (one of them is sample No. 2 in ref. [ 61, denoted now Ys2) having a large ( w 0.3 mm2) untwinned region in their centers, making possible the use of an electrical arrangement with the untwinned part between the voltage contacts. Although various groups have already published Au& results extracted from g(T) measurements in Y1Ba2Cu30,_b single crystals, [ 713 1, data of A&, Ax, and AIS=,extracted following consistent procedures from a(T) and x( 7’) measurements in the Same high-quality single-crystal samples, allow a more quantitative analysis of the OPF to be made, even up to reduced temperatures, e, of the order of 10m3,a region that may be expected

0921-4534/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.

258

A. Pomar et al. /Measurements ofparaconductivityin YBCO single crystals

to be well inside the so-called crossover or critical regions [ l-6,14- 16 1. In fact, the results of the present work confirm, to our knowledge for the first time at a quuntitatiue level, our early experimental results on the paraconductivity [ 5,141 and on the fluctuation-induced diamagnetism [ 51 obtained up to e~lO_~ in single-phase polycrystalline Y1BaZCu307_-6samples. Although so close to the transition we cannot exclude the influence of small stoichiometric inhomogeneities (which affect the critical temperature), these results strongly suggest that for e;52x10W2, both Au,(e) and A&(e) in Y1Ba2Cu307_-6compounds are affected by full critical order parameter fluctuations, in contrast with the mean field-like OPF’s that affect both observables for eZ2X 10-2. The second aim of this paper is to compare the amplitude and the temperature behavior of these Aa,( e) data not only with the previous fluctuationinduced diamagnetism data measured in the same samples [ 61, but also with the Ax and Atr data obtained in single-phase polycrystal samples [ 5,14,16,17], and with the existing theoretical approaches that take into account the double periodicity of the Cu02 superconducting planes in Y 1Ba2Cu307_-6 compounds [ 18-2 1 ] (in contrast with most of the existing AQ( e) analyses, where the conventional Lawrence-Doniach approach for one single periodicity is used [ 7- 121. These comparisons will allow us to obtain useful information on various open questions, in addition to what concerns, as noted before, the critical Aa, behavior and the reduced-temperature location and extension of the different OPF regions [ l-6,14-1 7 ] : the influence of the structural inhomogeneities at long length scales on the paraconductivity [22] and the effective number, N, (e ) , of superconducting ( CuOz ) layers in the unit-cell length [ 13,19-2 11. Also, we will obtain direct information on the two superconducting correlation lengths, in the ab plane and in the c direction. Finally, we will confirm two important conclusions obtained by analyzing our previous Aa,( e) and Ax&(c) data [ 56 1. The irrelevance in Y,Ba2Cu307_-d compounds of the pair-breaking (Maki-Thompson) effects on Acr,(e) [5,18,23,241, and that the number, n, of real components of the superconducting order parameter will be 2, i.e, the pairing state of copper-oxide superconductors will

be conventional &-wave pairing or one complex component unconventional pairing (extended or not &-wave pairing) [ 5,6,25 1.

2. Experimental details Crystals were grown in air from CuO-BaO-rich melts by using a previously described flux-growth technique [26,27], slightly modified [ 281. To this effect we mixed finely ground BaC03 (Aldrich 99.999%), CuO (Aldrich 99.999%) and decarbonated Y2O3 (Aldrich 99.999%) with a Y: Ba: CU molar ratio of 1: 9 : 23. The mixture was then compacted in pellets and sintered at 800’ C during 12 h. In the growing process, a pellet is placed into a yttria-stabilized zirconia crucible, heated in air in a box furnace to 970°C in 4 h and held at this temperature during 1.5 h. The temperature is then slowly reduced (at a rate of 5”C/h) to 870°C and, finally, to ambient temperature in 8 h. Crystals as large as 3 x 3 x 0.05 mm3 are easily identified in the solidified flux matrix, from which they must be mechanically extracted. In the postgrowth oxygen annealing, the crystals were placed in a boat within a tube furnace with O2 flowing. The furnace is then heated at 6OO”C, kept at this temperature for 2 h, cooled at 400°C in 1 h and held at this temperature during 4 days, in order to obtain good oxygenated 90 K single crystals [ 29 I. These last processes were repeated three or four times to improve both the normal-state properties (low pd and linear temperature dependence in the normal region) and to increase the superconducting transition temperature of the crystals. Crystals as described were examined under a polarizing optical microscope with white light, and characterized by standard singlecrystal X-ray diffraction with a ENRAF-NONIUS diffractometer, model CAD4, which allows us to determine the crystallographic axes for each sample. Most of the samples so obtained present large untwinned domains with ( 110) twinned microstructure at the edges. In some cases, the untwinned region is large enough and well centered so it is possible to make a four electrical contact arrangement along the a direction with the untwinned part between the voltage contacts (see below). This was the case for the two samples studied here, Ys2 and Ys3, the latter

259

A. Pomar et al. /Measurements ofparaconductivityin YBCO single ciystafs

being shown in the photograph of fig. 1. This sample has a quite regular platelet-like shape, with geometricaldimensions (inmm),d,=1.6, d,=0.5,dc=0.07, and with an untwinned region in its center of about 0.5 x0.5 mm2 (the region within the voltage contacts in fig. 1). The mass of this sample is 0.35 mg, and its resistive transition temperature (see below) is T,,= 90.8 K. The sample Ys2 was already used in the x(T) measurements published in ref. [ 61 (denoted in that work as no. 2). This sample also has a platelet-like shape, with dimensions (also in mm), d,=2.25, dp 1.2 and d,=O.O7, and it also presents an untwinned region in its center of 1.2x0.3 mm2. Its mass is 1.2 mg, and T,r=90.05 K. Other general characteristics of these two samples are described below and summarized in table 1. Before making the electrical contacts, high-precision measurements of x&e) (Hlab) and xc(e) (Hilab) are made by using a SQUID magnetometer (Quantum Design). A detailed description of the procedure followed and of the measurement accu-

racy, as well as the susceptibility results for the sample Ys2, may be found in ref. [ 61. Let us note here that as the mass of sample Ys3 is almost one third of that of sample Ys2, to have a similar resolution (of the order of 5%) the magnetic measurements of the Ys3 sample have been performed by using a magnetic field of @!J= 0.6 T, slightly higher than in the measurements of ref. [ 6 1. To check that the OPF effects in the ab plane are not affected by these magnetic fields and that, therefore, they may be compared with the paraconductivity measured in zero applied magnetic field, and also they may be estimated by using the conventional mean-field and scaling approaches for independent fluctuations of each order-parameter component, we may use the criterion that these approximations remain valid if [41,

(1) where 1, is the so-called magnetic length, e is the electron charge, fi is the reduced Planck constant; L(e)=&b(0)e-“2

(2)

is the superconducting correlation length in the ab plane (such a e dependence corresponds the MFR), and L(O) is the amplitude at T=O K; e=ln-

T Tco

T- Tco k: Tco ’

(3)

is the reduced temperature, and Tco and is the meanfield transition temperature. By combining eqs. ( 1) to (3 ), the above condition may be rewritten as = Fig. 1. Optical microscope photograph of the single crystal Ys3. The four leads for electrical resistivity measurements are also shown.

H H%(O)

(4)

,

where eo= h/2e is the flux quantum and HE$( 0) is

Table 1 General characteristics of the resistivity in the a direction, p., as a function of the temperature, of the two untwked YIBa2Cuj0,_6 single crystals measured in this work. The different resistivity parameters are defined in the main text. T, is the temperature where the magnetic susceptibility for the applied magnetic field parallel to the ab planes goes through zero Sample

~a(300 K) Wcm)

PJ~O’J K) Wcm)

P&OK) Wcm)

Q,/dT (pQcmK_‘)

Tl (K)

ATA (K)

AT; (K)

TC% (K)

Ys2 Ys3

214 227

61.5 70.0

-1.0 3.0

0.72 0.75

90.05 90.80

0.15 0.10

0.10 0.05

90.0 90.8

260

A.Pomar et al. /Measurements ofparaconductivity in YBCO single crystals

the amplitude (for T=O K) of the upper critical magnetic field parallel to the c direction. For M= 0.6 T, and if, for instance, Q( 0 ) = 1.1 nm, deviations to the zero-field OPF behavior may be expected if c 5 2 x 10B3, this last reduced temperature being less than the expected Ginzburg temperature (see later). Therefore, we may expect that for MsO.6 T, the OPF effects are not appreciably modified, in the whole studied e region, by the ag plied magnetic field used in our measurements. An important aspect of our present work is the approximation of the mean field-like critical temperature, T,, as the temperature at which & (for HIlab), that is not appreciably affected by OPF effects, [ 6 1, goes through zero. In fact, such a temperature agrees, to within the instrumental temperature resolution (0.1 K) , with the temperature where &( HI ab) goes also through zero. However, this temperature, that is going to be denoted Tcx, may be somewhat affected by an applied magnetic field of the order of 0.6 T. Such a decrease of T,(H) may be estimated by using the Abrikosov-Gorkov (AG) theory [ 30,3 11. In layered superconductors and for H perpendicular to the layers, the AG theory predicts ln T,(H) T,(o)

=&(0)rZ2

-

(5)

For mcO.6 T and TcO=90 K, and by using again Q(O)=l.l nm, wefound T,(O)-T,(H)k:O.l5K, a bit higher than the instrumental temperature resolution. The values of Tcx for the two samples studied here are given in the last column of table 1. These values have been obtained by using L(oH=O.2 T (sampleYs2) andM=0.6T (sampleYs3), inboth cases with Hlc, and they have been corrected for the corresponding magnetic field-induced temperature shift. After having performed the x(T) measurements, 50 pm diameter gold wires were attached to the samples in the ab plane, following a standard four-probe configuration. As we are interested in the intrinsic OPF effects in the superconducting Cu02 planes, to avoid the influence of the CuO chains, that modifies the electrical resistivity in the b direction, [ 9,321, the electrical contacts of the two samples were attached along the a direction, as illustrated for sample Ys3 in tig. 1. To this effect we first evaporated

through a mask four strips of silver, of 1 pm thickness and 200 ltm wide. Then, the gold wires were attached to the silver strips by using silver paste (Dupont conductor composition 4929) followed by heating at 400°C for 2 h in oxygen atmosphere. The final resistance obtained in such way was 50 mQ per contact, which is a medium-low value. As indicated before, the two voltage strips were placed within the untwinned region, in such a position that the measured resistivity corresponds to that of the a-axis. Note that the use of a conventional four-leads arrangement does not allow the determination of the resistivity in the other two directions, but it is much more direct and precise in order to obtain pa, the relevant resistivity to analyze the OPF effects. Finally, each sample was stuck with GE 7031 varnish on a sapphire substrate which was also varnished to a copper block, together with a calibrated Pt 100 thermometer. The longitudinal voltage was measured by using a conventional lock-in amplifier phase-sensitive technique (at 37 Hz ) described previously [ 141, the relative AC voltage resolution being better than 0.05 l.tV. The current densities (stabilized to better than 1%) were, typically, 1.5 A/cm’, which correspond to currents of the order of 0.5 mA. Thus, Joule heating in the current contacts was kept less than lo-* W. Correspondingly, as in our measurements the distances between the voltage leads were of the order of 0.5 mm, the (relative) resistivity resolutions were of the order of 0.5 @ cm. Let us stress here that we use as the voltage leads distance the one between the evaporated silver strips, which are very regular, in contrast, as can be seen in fg 1, with the silverpaste strips which do not make any good mechanical or electrical contact in the sample surface not covered by the evaporated silver. The main error source affecting the absolutep,(T) values is due to uncertainties on the sample geometry and electrical contact distances and it is estimated to be of the order of 10%. Each data point was measured after the electronic stabilization of the copper block temperature (with a Lake-Shore DRG91CA temperature controller), to within f 5 mK around the transition (89 Kc T< 130 K), and to within + 10 mK at higher temperatures.

261

A. Pomar et al. /Measurements ofparaconductivityin YBCO single crystals

3. Experimental results Figures 2(a), (b) and (c), 3(a) and (b) show the temperature dependence of pa, the electrical resistivity in the a direction, of the two samples studied in this paper. For clarity, only about 20% of the data points obtained has been plotted. The dashed lines were obtained by assuming a linear T dependence of pa in the normal region far from the transition, to avoid the OPF influence. More precisely, the &shed lines correspond to the fitting of p,J T) =paB( 0) + (dp,,/dT) T, with the temperature-independent coefficients p&( 0) and dp,.JdT as free parameters, in the normal (also called background) region bounded by 150 Kg Td 250 K. The adequacy of such a linear dependence is confirmed by the excellent quality of the fitting, the rms being 0.2% for Ys2, and 0.15% for Ys3. In turn, this linear temperature dependence of pJ T) is a first indication of the excellent stoichiometric quality of our samples. The corresponding background parameters are presented in table 1. Two comments are in order. Note, first, that these normal resistivity parameters are similar to within the 10% uncertainty for both samples. These +&/dT values also agree, to within the uncertainties, with those proposed before for untwinned YLBa2CZu307_-d single crystals [ 9,321. On the other side, the high quality of the linear fitting of p,a( T) in the normal region makes reasonable and accurate the use of its extrapolation through the transition as the background contribution to the measured resistivity around the transition. In figs. 2(c) and 3(b), we present a view ofp,( T) around the transition for each one of the two samples studied here, together with dpJdT. The temperature where dp./dT has its maximum is 90.80 K for Ys3 and 90.05 K for Ys2, and these are denoted T,,. As can be seen in table 1, these temperatures are similar, to within the experimental uncertainties (see below), to the corresponding Tcx, confirming our previous results [ 5,6] and suggesting that they must be close to Tco, the mean field-like critical temperature. Note also that these T,, values show that both samples are well oxygenated (6~ 0.1)) the small TcI differences between both samples probably just indicating a more difficult oxygenation of the sample Ys2, a relatively big single crystal. Another indication of the high quality of the two

200 z 6

150

2 d

100

50

0

0

50

100

150

200

250

300

T(K) 100,

,

,

1

\

_____-_. . __ 90

100

95

I

,

LD-like

105

4

110

T(K)

SO SAMPLE -----

Ys3

_------

----_

__----

60 -

s

G =I

40-

a” 20 -

0

cf.3 90

91

92

93

94

T (K) Fig. 2. The in-planelongitudinal resistivity along the a-axis of sample Ys3. The dashed line is the so-called background reaistivity obtained by fitting in the indicated background region. (b) Temperature dependence of the in-plane longitudinal resistivity around the transition. The dashed line is the normal-state background resistivity extrapolation through the transition. The dotted-dashed line is the mean field-like prediction in the indicated mean-field region and the solid line is the dynamic scaling prediction for the (full) critical region of the 3D-XY model. (c) Detailed view around the transition showing the location of the critical and mean-field regions and the brat fits to the theoretical appmachea in both regimes. Note that the presence of small compositional inhomogeneities (e.g., oxygen content) could also separate the p.( 7”) curve from the LD mean-field behavior for cS lo-*, without appreciably incnaing the width of the resistive transition.

262

A. Pomar et al. /Measurements of paraconductivityin YBCO singlecrystals

200 2

150

6 3 c”

100

50 n “0

50

150

-100

200

250

300

'I' (K)

90 RACKGROUND

SAMPUZ YrZ __---

OLW

a9

_________--------

90

91

92

93

T (Kl Fig. 3. The in-plane longitudinal resistivity along the a-axis of sample Ys2. The dashed line is the so-called background resistivity obtained by fitting in the indicated background region. (b) Detailed view around the transition showings the location of the critical and mean-field regions and the best fits to the theoretical approaches in both regimes. As for sample Ys3, the p.(T) data close to the transition (T- T,, S 2 K) could be affected by the presence in the sample of small compositional inhomogeneities. gee the main text for details.

YLBaZCu307_-6samples studied here is provided by the exceptional sharpness of their resistive transition, as can be seen in figs. 2 (c) and 3 (b). The half width of the resistive transition, AT,,, is defined by

this half width that will to some extent measure the uncertainty in T,, and, therefore, in the reduced temperature, e. So, the importance of having sharp resistive transitions is considerable, mainly to study OPF effects in the lower e region, closer to Tco. Naturally, the fact that in our samples TCxand Td agree to within AT& is also an important test on the validity of our choice of TCx as T,, as already commented on before. Let us stress here that the sharpness of the measured resistive transition does not exclude the presence of sample domains having relatively important differences in their transition temperatures, even bigger than the measured AT,. These differences could be due to, for instance, small differences in their oxygen content. In turn, these Td inhomogeneities could deform the p,( T) curves very near to the measured transition, in a similar way as the (full) critical effects (see next section) [ 221. In figs. 4 and 5 (a) we present the paraconductivity along the a direction, Acr,( e), of the two Y IBaZCu307_-6samples studied here. Acr,( e) is defined as 1 1 AU,(E) = - &J(C) /&zB(c)'

(7)

where po( E ) is the measured resistivity and pa ( E ) is the bare or background resistivity, i.e., the resistivity that the samples will have in the absence of OPF effects. These Ao,( e) data were extracted from the results of figs. 2 and 3 by using, as indicated before, T, as T,, in e, and for P&(e) through the transition of the linear extrapolation of p,( T) measured between 150 K and 250 K. The data of figs. 4 and 5 (a) are the central new

[I41

(6) where + or - corresponds to, respectively, the upper and lower half width. Their values for both samples are quoted also in table 1. In the case of the sample Ys3, with a quite narrow resistive transition, the difference between AT,, up and low is a direct manifestation of the OPF influence above T,,. In fact, as it is reasonable to suppose that the mean-field critical temperature, TCo,will be inside T,, ILAT:, it is

I. -3

I

-2.5

1

-2 Log,oi(T-~,,)

.

-1.5

a

i

-1

-0.5

/T,,I

Fig. 4. Paraconductivity along the a-axis of sample Ys2 vs. reduced temperature using three distinct values for T, See the main text for details.

A. Pomar et al. /Measurements ofparaconductivityin YBCO single crystals

T-T,,(K) 0.1

0.25

1

IO

2.5

25

5 :

&& ’

0.6 0.25

2

0.1

b 4

I

8 t

3

0.03

4 2 I -3

I

I

-2.5

-2

I

I

-1.5

-1

-0.5

Lo~,~~(T-T,,)/T,,~

-5,

I

-9’

a

’

-3

-2.5

-2

-1.5

’

-1

1 -0.5

Lo~,~~(T-T,,)/T,,~ Fig. 5. (a) Paraconductivity along the a-axis of sample Ys3 vs. reduced temperature. (b) Excess diamagnetism over T vs. reduced temperature measured with the applied magnetic field M=O.6 T, perpendicular to the ab plane. The 2&(t)/s abscissa assumes the e dependence in the MFR h(e) =&(O)C-‘/~ with &( 0) = 0.12 nm. See the main text for more details.

results of this paper. In particular, these figures provide, to our knowledge for the first time, quantitative information on Aa,( e), up to Ek: 1OS3, in untwinned Y ,Ba2Cu307 -a single crystals (previous paraconductivity data in untwinned Y1BaZCu307_-d single crystals extend only up to E= 10b2, and they focus on AQ values that combine p. and &, and that, therefore, complicate the OPF analysis [ 91). Various comments on these Aa, data are in order. Note tirst that the general c behavior of Au, not too close to T,, (for E2 10S2), is very similar to that of the paraconductivity that we have extracted from p(T) measurements in single-phase polycrystalline samples [ 5,14,33 1. Indeed, such a behavior is also similar (at least for lo-‘;5 e s 10-l) to that of the in-plane paraconductivity, Atsab(e), that may be extracted, following the same procedures, from the

263

&b(T) measurements in other Singk crystals published by other groups [ 7- 131. This result confirms then that the presence of structural inhomogeneities at long length scales (the characteristic length scales of the different structural inhomogeneities are much larger than the superconducting correlation length [ 61, even at Ek: 10b3) does not affect the e-behavior of the paraconductivity [ 5,14,33]. The main differences with our previous results in single-phase polycrystalline samples appear for e S 10m2, and they are mainly associated with a deficient estimation of Tco, due to the relatively larger transitions in polycrystalline samples (AT; 2 0.3 K). To illustrate the influence of the Tcochoice, in fig. 4 we have also represented Aa, by using TcI as mean-field critical temperature T,, T,, + 0.1 K and T,, - 0.1 K, utilizing a temperature shift that is close to the resistive transition half width, AT;, of sample Ys2. The positions of these three temperatures in the pa ( T) curve are indicated in fig. 3 (b) . As expected, and can be seen in fig. 4, these variations of Tco affect Aa, (6) only for reduced temperatures around AT; IT,, or lower, i.e., for e well below 10S2. Note that the influence of the critical-temperature uncertainties on the paraconductivity were earlier analyzed in ref. [ 17 1. In fact, it is very illustrative to compare our present paraconductivity results in an untwinned single crystal with those of fig. 3 of ref. [ 171: The e behaviors of the paraconductivity proposed in fig. 3(a) of ref. [ 171 and in fig. 5(a) ofthe present paper are very similar in the whole measured e range (up to 10S3), even at a quantitative level. The main progress of our present measurements in untwinned samples is, as stressed before, that due to the sharpness of the resistive transition, together with the x,(T) measurements, we may establish Tco much more accurately. In other words, our present data indicate to us that the correct choice among the three curves of fig. 3 of our 1988 paper [ 171, obtained from measurements in a polycrystalline Y1Ba2Cu30,_a sample, will be the curve of fig. 3 (a). We will see in the next section that such a choice has important implications on the intrinsic critical behaviors of the OPF effects in Y1BaZCu307_-dsuperconductors, As regards the amplitude of Aa,( e), note first that the results for both samples agree in the whole e range studied, to within the experimental resolution (of the order of lo%), as directly suggested by the corre-

264

A. Pomar et al. /Measurements ofpamconductivity in YBCO singlecrystals

spending dp,( T) /dT results. However, the amplitude of Ao,( e) is about 20% to 40% smaller, at least for e 2 lo-‘, than AQ,( e), extracted from our previous measurements in single-phase polycrystalline samples, or from the pa6( T) data in single crystal published by other groups [ 7- 13 1. By using the empirical ideas proposed in refs. [ 141 and [ 331 to take into account the influence on Au (and on p( T) ) of the long-scale inhomogeneities, it is easy to conclude that these differences are directly related to the fact that dpJdTk: 0.5 fl cm/K, i.e., typically 40% less than d,o@/dT.In turn, these differences are due to the influence of pb( T), which is affected by the CuO chains, on pob( T) [ 91. In the next section, we will see that these paraconductivity amplitude differences will rebound directly on the values that are extracted for the superconducting correlation length amplitudes, in the ab plane, Lb(O), and in the c direction, &( 0).

4. Comparison with the theory Independently of the detailed characteristics of the superconducting transition, one expects the existence of two different t regions above the transition, where the paraconductivity will have different critical behaviors [ l-4,1 5 1. A so-called mean-field region exists (MFR), not too close to the transition (see below), where the OPF effects are relatively small (compared to the conventional mean-field part, without fluctuations) and may then be approximated as a correction to the conventional mean-field behavior. On the other hand, there is a e region, closer to the transition, where, due to the divergence of the superconducting characteristic lengths and energies, the mean field-like behavior breaks down. In this eregion, called (full) critical, the contributions associated with the OPF effects become even more important than the mean-field (the values in the absence of fluctuations) part of each observable and, therefore, they cannot be approximated as a correction to these contributions. In addition, as noted before, this latter one cannot be calculated either by using a mean field-like approach. The reduced temperature separating the (full) critical and the meanfield e regions is the so-called G&burg temperature,

and for three-dimensional be expressed as [ l-41 1 ‘G=

%i?

(3D) fluctuations, may

1 2

kB &&(O)&(O)AC

’

(8)

where kB is the Boltzmann constant, and AC is the specific heat jump at the transition. By using for the YBa2Cu307_b system (see later, and ref. [ 6]), t&(0)=1.1 nm, &(O)=O.lZ nm, and [34]. AC=3.6x lo4 Jme3 K-l, we obtain eoa2x 10B2. Note that this eo value is consistent with the use of the 3D-Ginzburg criterion: by using the e dependence of & in the MPR, &(e)=&(o)e-1’2)

(9)

and again &(0)=0.12 nm, we obtain 2&(eo)=l.7 nm, a value larger than the interlayer Cu02 distances in the YBa2Cu30,_,r compounds (0.83 nm and 0.34 nm). Before summarizing the existing theoretical approaches for the direct (intrinsic) OPF effects in both regions (critical and MFR) indicated before, let us introduce already here some simplifying considerations that, as noted in section 1, are going to be confirmed by our present results. First, although the relevance of the non-intrinsic contributions to Aa associated with pair breaking (Maki-Thompson) effects is still controversial for the YBa2Cu30,_d compounds (which are in the clean limit) [ l&23,24], our previous Aa and Ax(r) results strongly suggest that these contributions are not appreciable in these compounds [ 5,6,14]: Not only Aa( e) may be fairly well accounted for (considering its amplitude and c behavior) by direct OPF effects but, more important, in the whole MFR the e behavior of Aa is very similar to that of A&, (for HI ab), this last observable not being affected by pair-breaking effects (see also later). So, in the present work the pair-breaking contributions to Aa are going to be ignored. Also, our previous AQ, and A& results [ 5,6] strongly suggest that the OPF effects are associated with a complex order parameter having two real components, i.e., n=2, as the one associated with conventional $,-wave pairing or one complex component unconventional (extended or non-&-) wave pairing. So, in the remaining part of this work we are going to assume n= 2. The ade-

265

A. Pomar et al, /Measurements ofparaconductivityin YBCO single crystals

quacy of this assumption will also be confirmed by the analysis of our present results in both the MFR and critical regions. The copper-oxide superconductors are layered materials, with superconducting CuOz planes coupled by Josephson tunneling. So, the best adapted approach to calculate Aa( e) in the MFR of these materials will be the Lawrence-Doniach (LD)-like extensions to layered materials of the Aslamazov-Larkin results for isotropic materials [ 35 1. In the absence of external fields (or in their weak limit), it is reasonable to suppose that the fluctuations do not interact, so each order parameter component will have independent fluctuations, which may be approximated as gaussians. In this gaussian approximation and for an order parameter having two real components, the LD-like approaches which take into account the double periodicity (two superconducting CuOz layers per unit-cell length) of he YBazCu90,_d compounds lead to [ 18-2 1]

(10) where AAL= &

~130 (Rem)-’

(11)

is the Aslamazov-Larkin conductivity for YBa2Cu30,_d compounds, with a unit-cell length of s= 1.17 nm, and BLD=

(2W) >2 s

amplitude and the e behavior of the paraconductivity in the MFR. To estimate iV,( e), we have calculated An=(e) in the gaussian fluctuation approximation on the basis of the time-dependent Ginzburg-Landau (TDGL) approach for a layered superconductor, with two different Josephson interlayer couplings, y1 and y2 [ 2 1I. Our procedure is very similar to that already used by Baraduc and Buzdin [ 201. The main difference in what concerns ZV,( e ) is that we have not imposed any given value to the relative coupling strength, y1/y2. The resulting ZV,( e ) is represented in fig. 6 as a function of c and for different values of the relative interlayer coupling strength. Note that y,/y2= 1 will correspond to the case where the Josephson coupling of each Cu02 layer with its two neighbor layers is the same; independently, for instance, of the different interlayer distances, or of the presence of CuO chains. The case where the coupling will be proportional to the square of the distance will be y,/y2= 6. As is confumed by a thorough analysis to be published in ref. [ 2 11, it is reasonable to suppose that in YBCO samples y1/y2 will take a value between these two limits. Therefore, for these compounds, we may approximate N,(c) by using yl/y2=l. In that case, N,(e) takes the very simple form, E-

1

I

,

, (s-1.17nm:~~o)-o.12nm,

(12)

is the Lawrence-Doniach parameter that controls the 2D-3D crossover of the dimensionality of the OPF in the MFR, and ZV,(e) is an effective number of independent fluctuating superconducting planes in a unit-cell length. In the YBa2CuJ0,_-d compounds, N,(e) may take v$ues between 1 and 2, and it will depend on BLD/e tkough the distinct strengths of the Josephson couplings between neighbor Cu02 planes [ 18-2 11. Note that N, = 1 will correspond to the conventional LD expression for Aa( e) [ I], currently used in most of the paraconductivity analyses in YBa2Cu,0, _d compounds [ 7- 12 ] although, as we will see below, such a simplification introduces in this case appreciable errors in the analysis of both the

Log,,KT-T‘,I,T,,I

1.2 -

-3

-1 -----

2

-2.5

-2 Log,,KT-T,,)

-1.5

-1

-0.5

i~,,l

Fig. 6. The effective number of independent fluctuating planes, N,(c), vs. reduced temperature, for different value8 of the relative interlayer coupliq stmngth. Inset: The mean-field exponent of AuJc) and &(c)/Tas a function of the reduced temperature. Two crossovers occur simultaneously: N,(c) varies from 1 to2andtheexponentof&~(~)and~(O/Tc~from-t to - 1. Below co the mean-field approach is expected to fail, and other exponents are in fact observed.

A. Pomar et al. /Measurementsofparaconductivity in YBCOsinglecrystals

266

(13)

Using eq. (13) in eq. (lo), we may rewrite AoJe) as (14) where AyF, the paraconductivity MPR, is defined by AMFE2A AL=260 0

(Rem)-’

and the paraconductivity cient is defined by BMF=4B 0

LD

*

,

amplitude in the

(15)

responding upper e limit (for the present calculations, not for the so-called mean-field region) will be around e k: 0.1. We will see that our present results confirm such an upper e limit for the LD-like paraconductivity without these various high c correo tions. In addition, for E> 0.1 the experimental uncertainties on Aa,( e), mainly those associated with the background estimations, become even larger than the measured A~J~(E). (2) By combining eq. ( 10) for Aa,( e) with our previous calculations for fluctuation-induced diamagnetism in the ab plane, r\;66 ( e) , in the MPR (see eq. (22) of ref. [ a]), we obtain (in MKS units)

2D-3D crossover coeffi-

A;$;;’ a (16)

We see, therefore, that in the approximation indicated above, Aa, in the MFR depends on two temperature-independent coefficients, Ay and BF”. Note also that, as illustrated by the results of Ne(e) given in fig. 6, in the 2D limit (&(O)e=s), i.e., if in the MFR B,+x E, then N,=2, and Aha, =2AAL e-i. This last expression is the conventional 2D AL result but with double amplitude. This just means that in the 2D limit the two planes in the unit-cell length fluctuate independently. In the 3Dlimit (&(O)=~~),i.e.,ifintheMFRBLp*e,then N,= 1, and An,(e) =AALB&/’ e-l/’ (which is the conventional 3D AL result), indicating that in this case the two planes in the unit-cell length are so strongly coupled that they contribute to the OPF fluctuations as an unique plane. Let us stress here that for any quantitative analysis of the paraconductivity of YBazCua07_d samples in the MFR in terms of the LD-like approaches, it is crucial to take into account the e dependence of N,, as can easily be concluded by just looking at the results of fig. 6. Let us, finally, stress here two aspects of these OPF effects in the MFR: ( 1) Whereas the lower t limit of applicability of the mean-field theories is, as noted before, the Ginzburg reduced temperature, eo, its upper e limit is associated with the slow-variation condition of the TDGL approaches [ l-41. Also, to obtain the above results we have not taken into account the possible influence of the local and high reduced-temperature effects. Our previous analysis suggested that the cor-

=2.79x

105efb(o) ,

(17)

a relationship that allows a direct extraction to be made of the mean-field Q( 0). Let us stress here that eq. ( 17) is independent of NC(e) arising in Ag,(e) and in Ah(e), and also of the number, n, of real components of the superconducting order parameters and, in fact, it was first obtained by us on the basis of the conventional LD approaches [ 51. Note also that in this equation, Aa, is assumed to be due only to direct OPF effects (see below). The dotted-dashed lines in figs. 2(b), (c), 3(b), 4 and 5 (a) are the best fits of eq. ( 14), with AyF and BvF as free parameters, in the e region 2x 10-2SeS 10-l. The rms fits are 3% and 4% for, respectively, sample Ys2 and Ys3. The resulting values are given in table 2. Various aspects of these results must be stressed: ( 1) These experimental Ay values agree, to within the uncertainties, together and with the theoretical Aslamazov-Larkin amplitude (eqs. ( 11) and ( 15 ) ) if the presence of two Cu02 superconducting layers per unit-cell length is taken into account. (2) The experimental By for each sample also agrees, to within the tmcertainties, to that of the other, and they take values slightly higher than the upper fitting limit (in a e region where, therefore, our meanfield approach is supposed to be still applicable), but almost one order of magnitude bigger than Ed. As the 3D-2D OPF crossover in the MFR is controlled by By, these results, confirming our previous results [ 5,141, indicate that in the MFR of YBa2Cus07_6 the OPF effects have a 3D-like behavior (in contrast, for instance, with the 2D-like behavior of the

267

A. Pomar et al. /Measurements of paraconductivityin YBCO single crystals

Table 2 Experimental values of the parameters arising in the theoretical descriptions of the paraconductivity in the a direction of the two untwinned YIBa2C~s07_-bsingle crystals studied here: The LD-like approach (by taking into account the double interlayer periodicity) in the MFR, and the scaling approach for the 3D-XY model for the (full) critical region. The corresponding fitting regions are also given Sample

AMP s (a-’ cm-‘)

Fitting region

Ys2 Ys3

Critical region

Mean-field region

cti

cut’

3x 10-Z 2x 10-s

10-l 10-l

BMF s

230f40 28Ok50

0.13+0.03 0.16kO.03

paraconductivity of the Bi-based HTSC’s, as first observed in our group [ 5,361). From these ByF values, and using eqs. ( 12) and ( 16), we obtain the amplitudes of the superconducting correlation length perpendicular to the CuOz planes, L(O), indicated in table 3. These values are slightly smaller than those obtained from our earlier AQ,( E) analysis [ 5 1, but this was based on the conventional LD approach, not well adapted to the YBa2Cu~0,_6 system, as noted before. These values agree to within the uncertainties, however, with those of scenario A of table 1 of ref. [ 6 1, which was obtained from a similar analysis (which takes into account the double CuOz periodicity ) in the MFR of our AQ( e) and Ax,J e) experimental results in YBaZCu307_-d single crystals (see also below). Another important aspect of these da,,(e) data in the MFR may be analyzed by comparing the results of figs. 4 and 5 (a) with the fluctuation-induced diamagnetism in the ab plane (HIab), Ax& E), measured in the same samples. The measured Axab(e) for sample Ys2 has already been published in ref. [ 6 1. Similar results for the Ys3 sample studied here are presented in fig. 5 (b). Note first that, in full agreement with the predictions of eq. ( 17), and with our Table 3 Mean-field (Ginzburg-Landau) correlation length amplitudes (at T=O K) and Ginzburg reduced temperature, co, deduced from the analysis of paraconductivity and of tluctuation-induced diamagnetism above Too Sample

Q(0)

Ys2 Ys3

l.Of0.2 1.1 f0.2

(mn)

&CO) (nm)

eo

0.11~0.02 0.12 + 0.02

(3f2)~10-~ (2f2)x 10-z

Fitting region t’ov

eUP

3x10-3 3x lo-)

3x10-s 2x10-z

critical exponent, x

Critical amplitude, AD, ( lo3 R-l cm-‘)

-0.33f0.05 -0.28f0.08

l.lSf0.20 1.20f0.25

previous experimental findings [ 5,6] the data of figs. 5(a) and (b) clearlysuggestthatAa,(e) and A&(c) have the same E behavior in the whole mean tieldlike region. This results is confirmed at a quantitative level, in fig. 7, where we present (Ab( e) / T) / Ag,( e) as a function of T- Tcofor sample Ys3. The dotted-dashed line in this figure was obtained from eq. ( 17 ), and the corresponding value of the amplitude, L6( 0)) of the mean-field superconducting correlation length in the ab plane is 1.1 nm. The corresponding value for the Ys2 sample is given in table 3, and it has been obtained by combining the data of fig. 4 with those of fig. 2(a) of ref. [6] for Aa( The errors of these Q( 0) values in table 3 correspond to the dispersion in the (AxJe)/T)/Ab,(e) data points. We may, therefore, propose here that from our paraconductivity and induced-diamagnetism measurements in the MFR in high-quality untwinned YBa2CuJ0,_-d single crystals, the amplitude (for T= 0 K) of the mean-field correlation length in

T-T,,(K)

Fig. 7. Relationship between the excess diamagnetism over Tfor H applied perpendicular to the ab plane and the excess conduo tivity in the a direction for sample Ys3 in the mean field-like region. The dotted-dashed line corresponds to the theoretical prediction for the OPF in the mean field-like region (eq. ( 17 ) ) , withQ(O)=l.l nm.

268

A. Pomar et al. /Measurements ofparaconductivityin YBCO single crystals

the ab plane of these HTSC compounds is &(0)=(1.1+0.2)nm,

(18)

a value somewhat smaller than the currently proposed amplitude [ 37 1, but one we consider very reliable. Let us stress also here that this Lb(O) value is about 15% bigger than that we have proposed previously from a similar analysis of AQ,(E) and Axa ( c ) data in single crystals [ 6 1. These differences are mainly due to the use of Atr& e), instead of Aa,( this last observable (not affected by the presence of the CuO chains) being, as noted before, the adequate one to compare with Ab( 6). Let us stress also here that these results in the MFR confirm then the absence of appreciable pair-breaking (MakiThompson) contributions to Aa,( and also the adequacy of n=2 (g= 1). An aspect of fig. 7 that is particularly interesting is that in the low-temperature side, i.e., for T- T&S 2 K, the experimental data separate from the temperature-independent behavior predicted by the LD-like approaches in the MFR. This result, already observed in fig. 4 of ref. [ 6 1, suggests that the appearance in this region is close to the transition of the critical OPF, instead of the mean-field OPF observed at higher temperature distances from the transition. This behavior is much better observed in the logarithmic representation of fig. 8. For ~52xlO-~, (Ax&e)/T)/Acr,(e) is not temperature independent, but proportional to e-‘13. In the

remaining part of this paper we will see that such a behavior of the paraconductivity near TCo in YBa2Cu30,_b can easily be understood in terms of a critical (non-mean-field) OPF, confirming at a quantitative level our early paraconductivity results [ 16,171. As earlier suggested by various theoretical [ 151 and experimental [ 16,171 works, the best adapted model for the criticalOPF effects in copperoxide superconductors is the 3D-XY model [38]. On the basis of such a model, the paraconductivity and the fluctuation-induced diamagnetism in the critical region may be written as [ 15,211 Aa,(c)=A:e-“‘,

(18)

Ax& e)/T=A;e -2’3 ,

(19)

where, taking into aCCOUnt their Scaling at eG with the mean-field expressions [ 5,6], it is easy to deduce [ 2 1 ] that the reduced temperature-independent amplitudes, A: and A;, are given by

(20) and ,+2,$(1+

2)-“2e~1/3.

(21)

In the above expression, As, the Schmidt diamagnetism (over 7’), is defined as [ 61, As

=

Pu&lcb(O)=3.63x 3&s

109&&(0) ,

(22)

where the right side of this equation corresponds to MKS units. By combining eqs. ( 18) to (21), we obtain for e 5 eo (23)

Fig. 8. Log-log representation of the relationship between the excess diamagnetism over T for H applied perpendicular to the ab plane and the excess conductivity in the a direction for sample Ys3 vs. the reduced temperature. The solid line is the dynamic scaling prediction for the (full) critical region of the 3D-XY model (eq. (25 ) ), with the corresponding parameters given in table 3. The dotted-dashed line is again the theoretical prediction for the OPF in the mean field-like region (eq. ( 17 ) ).

a relationship that, in contrast with eq. ( 17 ) for the mean-field regime, depends on e - ii3. We will see below that this result explains, even at a quantitative level, our experimental fmdings for e < I Oe2. by introducing the correlation amplitude, c&(e), in the critical regime [ 38 1, (24)

A. Pomar et al. /Measurements ofparaconductivityin YBCO single crystals

we may rewrite eq. (23) as (in MKS units)

(25) Note that the scaling at cd of Ca( to) and &,( eo) leads to &.(O) =&*(0)eti6

*

(26)

As for the YBa2Cu30,_b samples we have estimated eo=2x 10-2, &,(O) ~0.57 nm in these materials. We see, therefore, that (A&(e)/T)/Aa,(e) in the critical region diverges as e-l13, in contrast with the e-independent value for the MFR. This is, actually, the behavior observed in figs. 7 and, mainly, 8, for e < ed %2 X 10m2. The solid line in fig. 8 corresponds to eq. (25), with &(O) given by eq. (26), and with &JO)=l.l nmandeG=2X10-2.Theseresuhsconfirm at a quantitative level and in a very direct way, probably for the first time, our early proposal [5,14,16,17] that in the YBa2Cu30,_6 compounds the (full) critical regime is penetrated below T- T,;s2 K. These results confirm also the adequacy of the 3D-XY model (with n=2) to describe this critical behavior, as was early proposed [ 15 1. The excellent agreement between these results in the criticalregion and our experimental AU=(E)data in YBa2Cu30,_d single crystals is illustrated also in figs. 2(b), (c), 3(b), 4 and 5(a). The solid lines in these figures correspond to the best fit in the e region 3x 10-3
e
(27)

which is similar to eq. ( 18), but with A”, and x as free parameters. The resulting values are given in table 2, where we see that both the critical exponent and the amplitude agree to within the uncertainties with the theoretical values, x= - f and A: = ( 1.2 f 0.2) x lo3 (Q cm)-’ for both samples, these last values being obtained from eq. (20) by using the meanfield values obtained before (see table 3). For completeness, let us also note that the solid line in fig. 5(b) has been obtained from eq. ( 19) by using the same values than for Aa, ( e ) . As expected because of our results of ref. [ 61, the agreement with the experimental data is also excellent. However, let us stress here that these results on Aa,( e) and Ax&e) so close to the transition may be appreciably affected by sample inhomogeneities, as noted before in ref.

269

[22]. For instance, in the case of the electrical resistivity, the presence of very small oxygen differences between the borders and the inside of the samples could change somewhat the transition temperature of both sample regions. It is not difficult to see that small differences in the temperature transition, of the order of 0.5 K or less, does not appreciably affect p. ( T) for c 2 1OS2 but it could deform somewhat the measured resistivity for e 5 10e2, without appreciably enlarging the measured resistive transition [22]. So, to separate these possible inhomogeneity effects from the intrinsic behavior very close to TcI(c5 10m2) new measurements in samples with different annealing treatments will be very useful.

5. Conclusions This work on the paraconductivity in the YIBa2Cu307_-6 superconductors combines three basic ingredients: ( 1) The measurements are performed in highquality untwinned single crystals with exceptionally sharp resistive transitions (AT,: 50.1 K for sample Ys3), which allow us to obtain reliable information of the intrinsic paraconductivity in the a direction (not affected by the presence of the CuO chains) up to e2: 10-3. (2) These Aa.( e) results are compared with highprecision data of the fluctuation-induced diamagnetism measured in the same samples. (3) The theoretical approaches used in the MFR and in the critical region take into account the existence of two superconducting Cu02 planes in the unit cell. As a consequence, we believe that we are able to propose here a general scenario where the OPF effects on Aa,( e) and Ab( e) may be explained quantitatively and simultaneously: (i) The mean field-like region (MFR) will extend up to t k: 2 x 10m2.In this region both the amplitude and e behavior of AD,(e) (and of b(e), the fluetuation-induced diamagnetism for the applied magnetic field, H, perpendicular to the ab plane) may be explained in terms of the Lawrence-Doniach-like approach. The characteristic superconducting length amplitudesinthisMFRare,~(O)=(l.lf0.2)nm, and c(O)= (0.12f0.02) nm.

270

A. Pomar et al. /Measurements ofparaconductivityin YBCO single crystals

(ii) Below t=2x 10m2 the measured p=(T) separates from the mean-field behavior. Although so close to the transition we cannot exclude the influence of small inhomogeneities, these Au,(e) data for E5 2 X 1O-’ may be explained, also to a quantitative level, on the grounds of the 3D-XY model, with a critical exponent of x= - 4 for Au,(e), and x= - 3 for A&(e). In both e regions, it is crucial to take into account the presence of two Josephson-coupled Cu02 superconducting planes, their effective number, N,(e), varying between 1 and 1.6 for 10e3;5e5 10-l. This scenario corresponds to the scenario A of ref. [6] with one complex component order parameter i.e., conventional &wave pairing or one complex component unconventional pairing.

Acknowledgements

This work was supported by the CICYT, Spain, under Grant No. MAT92-0841. We thank J. Jegoudez and A. Revcolevschi for some of the highquality YBazCu907 _6 single crystals used in our preliminary measurements and for his hospitality during the stays of AD and FV in their laboratory of Orsay, France. FV wishes to acknowledge the hospitality of Juhen Bok during his stay in the l?cole Sup&ieure de Physique et Chimie de la Ville de Paris (ESPCI), as Invited Professor of the University Pierre et Marie Curie (Pairs VI), where the first version of this paper has been written. AD has benefited from an Ao cion Integrada Hispano-Francesa, No. 163B-1992, and from a Programa MIDAS No. 90/859 fellowship. We also thank I. Rasines and E. GutierrezPuebla, of the Instituto de Ciencia de Materiales de Madrid (CSIC) for the X-ray characterization of some of the single-crystal samples used now and in our previous OPF measurements.

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