Vortex fluctuations in BSCCO and YBCO

ELSEVIER Physica C 260 (1996) 41-51

PHYSICA

Vortex fluctuations in BSCCO and YBCO

V. Persico a V. Cataudella a F. Fontana b, p. Minnhagen a INFM-Dip. Scienze Fisiche, Universit6 di Napoli "Federico H", Napoli, Italy

b Dip. S.A.V_4., Universitf del Molise, Campobasso, Italy c Department of Theoretical Physics, Ume& University, Umett, Sweden

Received 18 January 1996

Abstract

The role of 2D vortex fluctuations has been investigated in BSCCO-2223 and YBCO films. Resistance versus temperature and current-voltage characteristics have been discussed in terms of the Ginzburg-Landau Coulomb gas model. It is shown that the analysis is able to extract new interesting information about the interplane coupling (2D-3D crossover) both for BSCCO-2223 and YBCO. Furthermore strong evidence that 2D vortex fluctuations play an important role for BSCCO-2223 is given.

1. Introduct ion

Vortex fluctuations play an important role both in low temperature superconducting films and in high T~ materials. In particular the resistivity tail mea- sured in the presence of transport currents and the nonlinear current-voltage characteristics are due to vortex core motion which produces dissipation.

In the last years the usual analysis of R - T and

I - V characteristics, based on the critical properties of Berezinskii-Thouless-Kosterl i tz (BTK) transition [1,2], has been criticized observing that, in many cases, the narrow temperature range in which the critical behaviour is expected cannot be properly resolved experimentally [3]. On the other hand it has been shown that the existence of vortex fluctuations of 2D character can be tested with the help of a scaling approach [4] which is not restricted to the critical region but only requires the Ginzburg- Landau Coulomb gas (GLCG) model to be valid [4].

The aim of this paper is to investigate the behaviour of BSCCO and YBCO samples for temperature less than the Ginzburg-Landau (GL) critical temperature To0 [5]. We will analyze our data (resistance versus temperature in zero magnetic field and current versus voltage) within the approach introduced in Ref. [4]. Our analysis allows to identify temperature ranges, above and below T c [5], where the data are well described by the 2D GLCG model. The temperature ranges are very different for the YBCO and BSCCO samples reflecting the different anisotropy of the two materials. Furthermore, even when the R - T data show significant deviations from the behaviour expected within the 2D GLCG model we find evidence that different samples of the same material behave in the same way showing universality within each material.

In Section 2 we describe the model on which the analysis is based. Section 3 gives the experimental details while our results are given and discussed in

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42 V. Persico et a l . / Physica C 260 (1996) 41-51

Section 4. We summarize and give some concluding remarks in Section 5.

2. The model

The role of two-dimensional vortex fluctuations in layered superconductors has been studied within the anisotropic 3D-XY model [7,8]. This is defined by the following Hamiltonian

/ xY = -J,{ E cos(0i- 0j) \ ( ij)

+3" I2 cos(0o-0o,)), (1) (nn') l

where the sums run over the nearest-neighbor (nn) pairs of a cubic lattice. Pairs in the same layer are indicated by (i j ) while, if in adjacent layers, by (n~).

This model assumes that the superconductor can be described within each layer n by the GL order parameter Sn = PfP-~-~ ei°n(r)- Furthermore we will as- sume that the superconducting mass density in each layer is uniform and takes the GL form Pn = P0( 1 - T/T~o) where To0 is the GL mean field critical temperature. In Eq. (1) we have JII = (h/mll)2Pn where mll is the mass of the superconducting particle in the layer and pn = t $~12 is the 2D superconducting mass density. The constant 3' = (mll//m ± )2, where m± is the mass in the direction perpendicular to the layers, introduces the anisotropy in one direction. The order parameter phase is also discretized within the layers; this is not expected to be a limitation as far as lengths larger than the lattice constant are of interest. The only dynamical variable in the model is then the order parameter phase 0i which provides a description of the dominant fluctuations, the vortices. In fact vortices can be viewed as phase configura- tions which contain branch lines since, within the GL theory, current and velocity are related to the gradient of the phase by

L = e* m*pnvn(r ) = e*p~hVO(r), (2)

where e* (m*) is the charge (mass) of the superfluid particle.

Both Monte Carlo simulations [7,9] and renormal- ization group approach [10] show that, although the

transition is in the universality class of the isotropic 3D-XY model, two-dimensional vortex fluctuations dominates the behaviour at temperatures slightly above the critical temperature. This result can be understood as a decoupling between vortices in different plane due to the vortices themselves [7]. It is worth commenting on the precise meaning of " two dimensional behaviour". Above a temperature/~ > T c where T c is the critical temperature of the model (the temperature at which the magnetization per spin vanishes) the vortex density shows a behaviour al- most indistinguishable from the one calculated in a pure 2D XY model [7]. As we will see, this gives a very useful indication on how to analyse resistivity versus temperature data in real high T~ superconductors. It might be noted that far from T~ a crossover from 3D to 2D behaviour for very anisotropic materials is always expected. However, the surprising result of Ref. [7] is that this crossover takes place at a temperature i~ > T~ extremely close to T~.

The results briefly reviewed above suggest that a systematic analysis of experimental data above T~ (resistance vs. temperature) and below T~ (current vs. voltage) and their comparison with analogous data for low T c thin superconducting films is of great interest.

In the case of 2D superconductors a good description is provided by the GLCG model [4]. In this 2D model the superconducting film, when vortices are absent, is described by the mean field solution of GL theory. The excitations dominating the fluctuations, i.e. vortices, are then added into this background mean field description. A vortex is in this description characterized as solution of the GL equations sub- jected to the constraint f ds . VO(r)= ___2qr where the line integral is over any closed path containing the vortex core. The GL equations give the vortex shape and the energy needed to create it [6]. The effect of thermally creating such vortices is taken into account within the Coulomb Gas (CG) model which describes a gas of thermally created vortices whose shape and formation energy are fixed and given by the GL equations. The system contains vortices of both + 1 and - 1 vorticity where the vorticity s is defined as s = (1 /27r ) f d s . VO(r). The interaction energy between two vortices is proportional to In(r) at large separation r and this allows us to consider the vortex gas as a 2D gas of

V. Persico et aL / Physica C 260 (1996) 41-51 43

charged particles [4]. The mapping between the two systems [4] gives the possibility to describe the vortex gas in terms of only two parameters, a) the CG temperature TCG

T

TcG(T) = 27rpo(T) (h /m , )2 , (3)

where T is the sample temperature and b) the GL coherence length ~(T) which defines the linear ex- tension of a charge in the Coulomb gas model. This mapping has been successfully used to find useful relations which do not depend on the details of the specific material under investigation or on the sample but, instead, only test to what extent the GLCG model is adequate for a correct description of a superconducting thin film [4]. In particular the model predicts that certain measurable quantities can only depend on the dimensionless variable X = Tcc(T)/Tc~(Tc), e.g.,

R / R N = f ( X ), a = h ( X ) , (4)

where R is the resistance below Tco, Rr~ is the resistance in the normal state, a is the nonlinear exponent in the current-voltage characteristic, V ot I a. In the case of low T~ superconducting films the relationships introduced in (4) have been tested showing that the GLCG approach describes correctly vortex fluctuation in those systems. It has also been Successfully employed to describe data for Y B C O / PBCO superlattice structures [11,12]. We want to emphasize that the above analysis is not restricted to the critical region and, therefore, is not directly related to the existence of a BKT transition. This is a crucial point in the present context since, although a layered superconductor has a 3D phase transition, 2D vortex fluctuations nevertheless dominate the behaviour in the immediate vicinity of the transition.

surements (details of deposition procedure along with the structural and superconducting characterization are reported in Ref. [15]). The samples (1200 ,~ thick) showed c-axis, epitaxial growth and good superconducting properties (T~ > 92 K, Jc(77 K ) > 107 A/ /cm 2, ATlo%_90 ~ <: 1 K).

BSCCO-2223 phase samples were produced by a novel technique based on chemical transport in thermal gradient enhanced KC1 flux [16]. The samples obtained were free standing platelets, about 1.0 txm thick and with a surface area of several mm 2. X-ray diffraction, microprobe analysis and transport measurements showed c-axis oriented samples with low normal state residual resistance and sharp superconducting transition.

R - T and I - V measurements were performed on YBCO samples (in particular, I -V characteristics were obtained on YBCO sample which was pat- temed by standard micrometric photolitographic technique to obtain a 20 ~m wide, 1 mm long line), while R - T measurements were performed on BSCCO ones. All measurements were made by the standard four-probe technique (DC for R - T measurements, pulsed for I -V characteristics to avoid self heating problems), the external magnetic field being shielded by a b-metal cylinder. All instru- ments were connected to an HP computer by a IEEE-488 bus-interface, and the whole measurement and acquisition processes were controlled by soft- ware.

In order to carry out accurate measurements in the temperature region around T~0 and to avoid unde- sired thermal fluctuations of the sample temperature, an original experimental setup, based on a pressuriz- able liquid argon bath, was used [17]. The temperature sensitivity achieved was better than 5 mK for R - T measurements, while the temperature stability was better than 20 mK for I - V measurements.

3. Experimental

High quality YBCO (YBa2Cu307_ x) thin film were produced by inverted cylindrical magnetron sputtering (ICMS) technique. The films were de- posited onto different substrates (MgO, LaA103) and characterized by scanning electron microscopy (SEM), X-ray diffraction (XRD) and transport mea-

4. Results and discussion

4.1. r>_rc

In Fig. la (BSCCO) and Fig. lb (YBCO) typical measurements of DC electrical resistance in the temperature region around T~0 are reported. In order to plot the ratio R / R N as a function of the dimension-


I I I I

o.s (a)

0.4

0.3

0.2

0.1

o.o , I , l , I , I ,

100 105 110 115 120 125

T e m p e r a t u r e OK)

t i . t , i , i 0.3 (b)

~ 0.1

O0 • I i I i I , I

91 92 93 94

T e m p e r a t u r e (K)

Fig. 1. (a) The ratio of resistance to the normal state resistance (R/R N) as a function of temperature for one of the BSCCO films (sample #1) . (b) The ratio of resistance to the normal state resistance (R/R N) as a function of temperature for one of the YBCO films (sample #1);

less variable X (Eq. (4)) which allows a systematic comparison of the data with those of low T~ superconducting thin films we proceed in the following way.

In order to avoid a many-parameter fitting procedure, we started with the assumption that the data

follow the universal 2D scaling curve (see Ref. [13] for an accurate definition) and checked the validity of this assumption a posteriori. From this assumption we were able to extract the values of the universal variable X by experimental R / R N data by means of numerical inversion of Eq. (1) of Ref. [13]. Defining z = In(R/RN) the inversion formula obtained was:

X = 1.5816z -2 - 2.2490z - I + 1.4984 + 0.0832 z

+0 .0029z 2 i f - 1.32> z > - 10.02,

1.143 X = 0.98 i f - 10.02 > z > - 14.35.

z + 4.824 (5)

Once the values of X were obtained, we checked our assumption and plotted T / X versus T. From the definition of X it follows that in a GL treatment of the superfluid density the ratio T / X should be a linear function of T, i.e.

T Po(T) T X T~ Po(r---- ~ c t l - T~---~" (6)

Figs. 2a and 2b show such plots in the case of a BSCCO and a YBCO sample, respectively. From the figures the following evidence can be extracted: 1) For our BSCCO samples the linear behaviour signaling 2D vortex fluctuations is borne out to very high degree. This is evident from the solid line in the inset of Fig. 2a which represents the linear best fit to the data (obtained by a least-squares procedure on a set of more than one hundred experimental points). The crossing of this line with the T-axis gives To0. T~ D is the crossing point between the solid line and the line T / X = T (broken line in Fig. 2a, compare Eq. (6)). This gives T~o = 115.19 and Tc 2D = 99.58. From this we conclude that the behaviour for BSCCO is dominated by 2D vortex fluctuations and also that we can in case of our BSCCO determine T¢o and T~ ° very well by this procedure. 2) The true phase transition is 3D and thus the true phase transition temperature T 3D is higher than the 2D phase transition temperature T~ ° implied by the 2D vortex fluctuations. In Fig. 2a the crossover to the 3D transition is reflected by the small deviation from the solid line at the lowest temperatures. The fact that the data falls slightly above the solid line at the lowest temperatures means that the vortex fluctuations are suppressed and thus that T~ D < T~ 3D.

V. Persico et a l . /Physica C 260 (1996) 41-51 45

100

80

60

40

20

0 100

' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' '

125 ~.

0 l e ° - 1 ~ [ ~ ' [ .........................................

~ 1~' ~ ' ~ rio ~

\ 1 (a)

105 110 115 120 Temperature (K)

100 I ISO 80 ~ so ~ ~

60 4°~~llm~ 4O

20 ~ o

(b) 0 , I ~ I , I 91 92 93 94 95

Temperature ( K )

Fig. 2. (a) The R - T data of Fig. la plotted as T / X versus the temperature T. X is obtained from the experimental R / R N data by means of Eq. (5). The straight line is a linear fit over the data having neglected the high temperature tail. The intersection between the straight line and the T-axis gives Too. See Section 4 for a discussion of this construction. In the inset is shown the intersection between the linear fit and the line T / X = T, T~ D. b) The same plot as in (a) for the YBCO film of l(b); here the linear fit over the data has been done keeping the intersection with the T-axis, To0, fixed. The value of To0 is taken from the I - V measurements of the same sample (see text for the details).

3) Also at the highest temperatures a small deviation from the linear behaviour is observed. This deviation we attribute to the amplitude fluctuations of the order parameter which is not taken into account inside the GLCG model. Consequently we will ne- glect this small region in our analysis. 4) The data for our YBCO sample in Fig. 2b is somewhat different. It is not possible to find a unique straight line signaling 2D vortex fluctuations is this case. However, as we will come to, it is possible to determine T~0 from the I - V measurements. From these measurements we have T~0-- 94.77 K for the sample in Fig. 2b. The solid line in Fig. 2b represents the best fit to the data for an a priori given To0 and this line gives T~ D-- 88,60 (compare Fig. 5a). Our interpretation is then that only the data for higher temperatures (over roughly a 0.5 K region) can be attributed to 2D vortex fluctuations in case of YBCO. For lower temperatures these fluctuations are strongly suppressed. Thus our con- clusion is that the resistance data for BSCCO and YBCO are qualitatively similar but quantitatively different. 5) We attribute the 2D vortex fluctuations to the 2D superconducting planes in the materials and the deviations from the 2D behaviour for lower temperatures to the interplane coupling. Thus we attribute the larger deviation in case of YBCO to a stronger interplane coupling. The interplane coupling is in- deed very different for these two materials [19], ranging in energy units from 10 K for YBCO to 0.1 K for BSCCO-2212. According to this interpretation, the deviations represent a measure of the interplane coupling and signals a crossover from 2D to 3D behaviour.

The crossover mentioned above is expected to be observed in the plot R / R N versus X, provided that the values of X obtained by the linear fitting are used. The results are plotted in Fig. 3, where experimental data relative to two different samples of YBCO and to two different samples of BSCCO are reported. The solid line represents a parametrization of the universal 2D scaling curve f ( X ) 13 (compare eq. (4)). The agreement among the data and the universal curve is quite striking. This allows us to attribute 2D vortex fluctuations as the source of the finite dissipation in a region below T~0, both for BSCCO and YBCO. For BSCCO a deviation from the 2D curve is observed only close to Tc 2D ( X < 1.3).

46 V. Persico et aL / Physica C 260 (1996) 41-51

0 i ! I i i

-2

-4

,~ n YBCO (Sample #1)

- 8 /g o BSCC? !_Samp!¢ # ! ?

/ O o ~ BSCCO (Sample #2) -10 [ ~ 2D Universal Curve

I o ra

-12 , i , m , i , i , , , 1.0 1.5 2.0 2.5 3.0 3.5 4.0

X

Fig. 3. The logarithm of the resistance data R / R N as a function of the scaling variable X for all four samples. At each temperature X = T(Tco - T)D)/[T2D(Tco -- T)] is calculated by using the values Tco and T 2D extracted by the same type of construction reported in Fig. 2a and Fig. 2b; in the plot is also reported for comparison (full line) the universal resistance curve for 2D superconducting films (see tex0.

For YBCO the deviation is comes somewhat earlier (X < 2.1). In the picture discussed above, this deviation has been attributed to the interplane coupling which leads to a crossover of the "effective" dimen-

sionality of the system. The crossover point corresponds to larger R / R N the stronger the coupling is. In this sense, our data confirm the observations that YBCO is less anisotropic (i.e., with stronger values

7

6

5

O

J

.~ 3 o~

2

D []

[]

O

10 4 i " . . . . . . . . . . . . . . . " " ' " " ' ' " " ' ,ol °E/////

O

[]

0

i 0 -s 10 .4 10 "3 10 -z 10-i 10 o 101

Cnrrent (mA)

13 0

O

[]

o

I nOD [] 0

, I I I , I i I ,

88 8 9 90 91 92 93

T e m p e r a t u r e ( K )

Fig. 4. The nonlinear exponent of current voltage characteristics (V ~ I a(r)) as a function of temperature for the YBCO (sample #1). In the inset are reported the bare ( I - V ) data from which a(T) has been extracted.

V. Persico et a l . / Physica C 260 (1996) 41-51 47

of J±) than BSCCO. Similar conclusions have been reached in the case of YBCO/PBCO-superlattice structures [11,12].

It is interesting to note that the data for both our YBCO samples in Fig. 3 follow the same curve when plotted against the scaling variable X. This

indicates that the deviation from the 2D universal curve is a property of the material i.e. as long as the YBCO material in the two samples are identical we expect their properties to be identical. However, this is not the complete truth because the two YBCO samples are slightly different. One is characterized

[ -

soo [

600

400

200

(a)

88 90 92 94 96

Temperature (K)

30

25

20

+ 15

10

750

500

v - ~

250

0 2.5

I

(b)

0 82 94

,l,l,I,l.~liT 84 86 88 90 92

T e m

' ' • • I . . . . 1 , • ° ,

(c)

T,/,,

] _. . . . . . . . . .

_ _ - - ' - - " W

_ _ - - - - V

| ! •

3.0 3.5 4.0

Temperature (I~

~erature

Fig. 5. (a) Nonlinear l - V data from YBCO (sample #1) . The data points are plotted as (a + 1)T against temperature T. The straight full line is a fit over the low temperature data. The intersection between the straight line and the T-axis gives To0. The dashed line is (a + 1)T = 4T. (b) The same construction (a + 1)T against temperature T as in (a) for a single crystal BSCCO-2212 sample from Ref. [22]. (c) The same construction (a + 1)T against temperature T as in (a) for a low T e film (Xe-Hg alloy from Ref. [21]).


by Tc0 = 94.77 and T 2D = 88.60 whereas the other is characterized by T~o ~- 92.12 a n d Tc 2D ~ 87.30. Nev- ertheless when plotted against the scaling variable X the resistance data is identical. This suggests that the interplane coupling which causes the deviation from the 2D universal curve is to large extent a function of the scaling variable X.

4.2. T<_T c

For 2D superconductors in the temperature region T_< T c, vortex-antivortex pairs are bound so that no free vortices are present in the system and no dissipation can occur. Nevertheless, in the presence of small currents flowing into the sample some of the pairs can decouple leading to the excitation of free vortices and associated dissipation. The voltage drop produced (and the subsequent electrical resistance) is related to the applied current field by a power law:

V C~ I a(T),

where the exponents a(T) is directly related to the 2D superfluid density, which exhibits a universal jump at T c [18] A very similar scenario has also been proposed for layered superconductors with the only difference that, now, a threshold current is expected [20]. Since, as for T > T~, the 2D GLCG model predicts the existence of universal behaviour (see Eq. (4)) for the exponent a, we are interested in testing

to what extent the 2D GLCG is applicable for T < T c in the case of layered superconductors.

I - V characteristics measurements have been performed on the YBCO samples. In the inset of Fig. 4 a set of measurements made for T < T~ has been reported (the data is for the same sample as the resistance data shown in Fig. 2a. The data is shown on a log-log plot in order to make the power law behaviour more evident. Fig. 4 shows the temperature dependence of the exponent a(T) extracted from the data.

Within the GLCG model for two-dimensional and layered superconductors the value of a is related to the reduced temperature X by [14]

4 e ( l ) a ( X ) - 1, (7) x,(x)

where E(X) is the dielectric constant which describe the screening due to bound vortex-antivortex pairs in CG model. Eq. (7) has been proposed recently in ref. [14] on the basis of scaling arguments. In particular we are interested to compare our data to a CG model with fugacity z = e -Eo/¢r"r) = 0 where Ec is the non-electrostatic energy needed to create a vortex (core energy). Any CG model will reach the limit z ~ 0 for X << 1 and in this limit E(X)- - 1 because the screening due to thermally created vortex-antivortex pairs vanishes in this limit. In the z = 0-limit

10

6

. . . . i . . . . i . . . . i . . . . |

t ta YBCO (Sample #1)

\ I v Hg-xo l /

. . . . . . . . . . . . . .

1 1

ta ~ v I:1 0

[ ]

V v

O

2 t~D D ° o

i i i i I i i i i I i i i , I , , , ,

0.10 0.15 0.20 0.25 0.30

Fig. 6. Coulomb gas scaling for nonlinear I - V exponents. The exponents (a + t) are plotted against the scaling variable .~ for the samples of Fig. 4. For comparison the curve a + 1 = 1/3~ has been plotted (full line). See Section 4.2 for the details.

V. Persico et a l . / Physica C 260 (1996) 41-51 49

the BTK transition temperature is given by 4TCGe(T cG) = 4T cG = 1, i.e., T cG = ±,. Conse- quently X takes the f o r m

1 1 _ _ r L0 - T, / , (8) ~ X ( z = O ) = X = - ~ T1/4 T~o-T '

where Zl/4 is defined as the temperature at which ~" = 1. In order to extract T~0 from our data below T~ we note that

1 T - Tc0 ( a + 1) =- -z c t - - (9)

X T

and for low enough temperatures. In Fig. 5a we have plotted (a + 1)T vs. T for the same data as in Fig. 4. For low T the data is linear in accordance with Eq. (9). The solid line in Fig. 5a is the best fit to this linear regime. The crossing point with the T-axis gives To0 = 94.77 K. We used this determination of T~0 for the resistance data in fig. 2b. The broken straight line in Fig. 5a is given by (a + 1)T= 4T and the crossing point between the broken line and the solid line gives T1/4 -- 91.47 K. By aid of Eq. (8) we can now plot a + 1 versus X and this is shown in Fig. 6 (solid squares). The solid curve in Fig. 6 corresponds to the limit of vanishing screening ef- fects due to vortex-antivortex pairs ~ ( X ) = 1 so that this curve is just 1/~' . The data follows the solid curve for small X. However, as the transition is approached the vortex fluctuations come into play and e (~ ' ) increases.

If the exponent a is associated with 2D vortex fluctuations, then a = 3 precisely at the corresponding BKT transition [18,14]. This criteria gives the estimate T~D= 90.51 K. Now from the resistance data we obtain T~ D ~ 88.60 which is clearly smaller. The difference, according to our interpretation, is that the 2D vortex fluctuations for the resistance is associated with the individual planes in the layered material due to the vortex driven plane decoupling above T~ D. Below T 3D there is on the other hand no such dramatic decoupling and the interplane coupling is fairly strong for the YBCO material. Thus we associate the 2D vortex fluctuations for our (1200 /~ thick) YBCO sample with vortex fluctuations of the sample coupled together to a single (1200 thick) film. Thus the vortices in this case goes through the complete sample. The BKT transition for

the collectively coupled planes is higher [11] and consequently this interpretation explains the difference between T~ D characterizing vortex fluctuations above and below T c.

In order to get some further insight we have compared our a(.~) results for the (1200 ~ thick) YBCO sample with a low T~ film (Xe-Hg alloy from Ref. [21]) and for a single crystal BSCCO-2212 sample from Ref. [22]. Fig. 5c shows the (a + 1)T vs. T construction for the low T e film. The construction is the same as in Fig. 5a. However , in the present case T~o was well determined from the resistance data (T~o = 3.88 K [23]). The solid line is the best fit to the data with the constraint that the crossing with the T-axis is at To0 = 3.88 K. The crossing between the solid line and the broken line gives 7"1/4 (~ 3.35) as in Fig. 5a. In parallel with Fig. 5a we can now plot the data as a vs 3~. This is given by the triangles in Fig. 6. Fig. 5b gives the same construction for the BSCCO sample of Ref. [22]. The solid line is the best fit to the linear part of the data and this line determines both Tc0(= 92.61) and T1/4(~ 86.74). The corresponding a (X) is plotted as open diamond in Fig. 6. The broken horizontal line in Fig. 6 is a + 1 = 4. This line corresponds to the BKT condition and consequently only the data with a + 1 > 4 can be associated with 2D vortex fluctuations in the low temperature phase. We will only attempt to explain this part of the data. First of all one notices that the low T~ data and the BSCCO data when plotted in this way is strikingly similar (all the way up to a + 1 = 4). This w e interpret as evidence of 2D vortex fluctuations described by a universal a (X) associated with the GLCG model. Since the BSCCO sample is a 3D single crystal, we conclude that the 2D vortex fluctuations in this case have to be associated with individual planes. This interpretation is consistent both we the fact that the interplane coupling for BSCCO is very small and with the strong manifestation of 2D vortex fluctuations in our resistance measurements. It is tempting to press the interpretation slightly further and argue that the small interplane coupling which nevertheless exists will tend to suppress 2D vortex fluctuations associated with single planes. This will make e (X) slightly smaller than for the pure 2D case. Conse- quently a (X) should be slightly larger (compare eq. (7)). This is consistent with the a + 1 > 4-data in


Fig. 6. The a(A')-data for our YBCO sample by comparison suggest a large increase in E(3~) as X is increased. In this case we associate the 2D vortex fluctuations with the planes collectively coupled together into a single (1200 .~ thick) film. As the transition is approached vortex loops associated with individual superconducting planes are spontaneously created. These additional fluctuations decrease the superfluid density associated with the collectively coupled single film. A decrease in the superfluid density corresponds to a decrease in a, since the superfluid density is proportional to 1/e(3~). The large decrease of a(X') compared to the low T~ data is consistent with this interpretation.

5. Concluding remarks

ear/-V-characteristics. We concluded that these 2D vortex fluctuations were associated with vortices penetrating the whole sample coupled together to a single 2D superconducting film. The nonlinear I - V exponent a showed a crossover from a regime controlled by vortex fluctuations made by vortex lines going through the film and a regime where vortex loops fluctuations become important. We compared this to data for a BSCCO single crystal and concluded that also in this case the nonlinear 1-V-characteristics was due to 2D vortex fluctuations but this time associated with the individual superconducting planes in the material.

It should be clear from the present measurements and the analysis of them that new interesting information concerning the high-T~ superconductors can be extracted from resistance and I - V measurements.

Our analysis showed that the resistance for our BSCCO samples was due to 2D vortex fluctuations over a large region. It furthermore showed that these 2D vortex fluctuations were well described by the 2D GLCG model and the corresponding 2D universal resistance curve. To be more precise we showed that, when plotted against the scaling variable X, the resistance ratio R / R N follows the 2D universal curve over roughly three decades. We associated these 2D vortex fluctuations with the individual superconducting planes due to the CuO 2 layers. For the smallest resistance ratios there is a deviation. This we associated with the interplane coupling which suppresses the 2D vortex fluctuations and drives the transition from being 2D of BKT type into a 3D transition.

Also for our YBCO samples we concluded that a part of the resistance could be ascribed to 2D vortex fluctuations. However, in this case the interplane coupling is much stronger and the deviation from the 2D universal curve comes at much higher resistance ratios. Interestingly enough the resistance ratios for our two YBCO samples follow the same curve when plotted against the scaling variable X. This suggests that the deviation from the 2D universal curve is a property of the material and that furthermore this material property is only a function of the scaling variable X. According to our interpretation this property is associated with the interplane coupling.

Our YBCO samples also showed evidence of 2D vortex fluctuation below the transition in the nonlin-

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Vortex fluctuations in BSCCO and YBCO

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