Detection of the flux-line lattice in high-Tc superconductors by ESR-probe decoration

Physica C 174 ( 199 ! ) 447-454 North-Holland

Detection of the flux-line lattice in high-T superconductors by ESRprobe decoration Y u . N . S h v a c h k o , A.A. K o s h t a , A.A. R o m a n y u k h a a n d V.V. U s t i n o v Institute of Metal Physics, UralDivision of the USSR Academy of Sciences, 620219 Sverdiovsk, USSR A.I. A k i m o v Institute of Solid State Physics and Semiconductors, Byelorussian Academy of Sciences, 220726 Minsk, USSR Received 7 December 1990

ESR measurements on YBa2Cu307 and Tl2Ba2Ca2Cu3Oto cerar, .ic samplescoated with the organicradical DPPH are presented. The flux-line lattice (FLL) detected by inhomogeneous line broadening is used to estimate the magnetic penetration depth ~.(0) = 3500 A for YBa2Cu307 and 3900 A for TI2Ba2Ca2Cu3Oio. The extracted temperature dependences 2(T) are consistent with the expected ( ! - ( T/To)4 ) - t/2 behavior for both compounds investigated.

1. Introduction

Since the discovery of the novel high-To compounds there have been performed a lot of experiments aimed at obtaining exact values of superconducting parameters, in particular the magnetic penetration depth 2(T). However, the extracted data were strongly dependent on sample preparation and on the technique used. Experiments show a variety of estimated values ~(T): 2500-6500 A for L a - S r - C u - O [ 1,2,3 ]; 2008000 A for Y - B a - C u - O [4-9,10,11 ]; 3500 A for BiS r - C a - C u - O [12] and 1700 A for T I - B a - C u - O [13] depending on the magnetic field orientation with respect to the crystal. Most of the locally sensitive techniques used are based on FLL detection. These include ~t+SR, NMR and polarized neutron scattering [ 14-18 ]. Other methods (DC magnetization [ I I ] and Ac, . . . . . . . . :~,:1:,,, -7 1o ] ) , , ,, pend on the distribution of FLs over the sample investigated. It has been shown recently [20] that 2-values obtained from ~t÷SR or NMR should be modified by taking into account the real field density distribution, n ( B ) . It appears that the n ( B ) function [22] influenced by the pinning effect [ 20] demonstrates

a peculiar asymmetric shape, in contrast to the gaussian local field distribution underlying the evaluation of 2. However, the modified treatment of~t+SR data gives either values too large for 2 ( 0 ) without pinning effects or too small for the case of strong pinning. Thus, certain difficulties connected with the correct calculation of FLL [20,21] arise for the interpretation of both ~a+ SR and NMR experiments. in this paper we develop the organic-radical coating method first suggested by Rakvin et al. [23].

2. Experimental method and samples

The ESR spectra were recorded employing standard homodyne ERS-230 and ERS-231 X-band spectrometers. The spectra were obtained in digital form by employing a computer-controlled slavemaster, which made it possible to accumulate signal data in the course of repeated passage through the resonance conditions, and also to determine the parameters of the ESR line. The temperature of the sample was set and stabilized by an Oxford Instruments ESR 900 continuous-flow helium cryostat, in which temperatures could be maintained in the range 3.5-300 K to within 0.1 K. The temperature of a

092 !-4534/9 !/$03.50 © i 991 - Elsevier Science Publishers B.V. ( North-Holland )

Yu.N. Shvachko et al. / Detection of the flux-line lattice by ESR probe

448

Table I Parameters of the samples investigated

20

Parameter

YBa2CusO7

TI2La2Ca2CusO~o

a (A) c (A) T~ (K) ATe (K) X" 104 (emu/mol) P l n (mf~ cm)

3.823 1 !.670 94 2 3.0 0.40

3.851 35.621 122 10 4) 3.1 3-8

-

.da

r•16

4) A small tail in the temperature dependence of static magnetic susceptibility was observed down to about 100 K.

sample was monitored using an AuFe-Chromel thermocouple. All investigations were carried out on ceramic samples of YBa2Cu307 and Tl2Ba2Ca2Cu30:o. The lattice parameters (a, c), superconducting transition temperatures To(onset), the transition width ATe measured between 0.1 and 0.9 levels of AC susceptibility (real part, v= 107 Hz), and some other parameters are presented in table 1. Ceramic YBa2Cu307 samples were synthesized at the Institute of Metal Physics, Ural Division of the USSR Academy of Sciences. Tl2Ba2Ca2CuaO,o samples were synthesized at the Institute for Solid State Physics and Semiconductors, Byelorussian Academy of Sciences. Paramagnetic centers of an organic radical, DPPH (diphenyl-picrylhydrazyl) with rather weak spin-lattice relaxation (AH(300 K) ~< 2 0 e ) were deposited on the grain surface from the solution in acetone or as fine-particle powder.

3. Experimental results

The temperature dependence of the ESR linewidth for the pure DPPH powder that was used (see fig. l ) appeared to be weak down to T= 20 K with AB(300 K ) = !.2 G and might be fitted by the in= verse temperature behavior A B ( T ) = A B ( 3 0 0 K)/ (T/300) I/2 over the entire range. The corresponding behavior of Hr is nearly temperature independent, with the value Br(300K) =3371.6 G down to the liquid helium range, giving the g-value 2.0043. Therefore, this kind of ESR-probe seems to be convenient for the investigation of field inhomogeneity

~

E

F,~ 4

l

~

,

,

t

,

)

i I °°°

O0

,

,

8_ _ I

lOO

,

& _ _g I

200

I

I

300

Temperature, K Fig. 1. ESR linewidth, AB(T), measured in ceramic samples YBa2Cu307 (squares) and in pure DPPH grains deposited on a mica substrate (circles). The solid line is the approximation with the use ofeq. (2) and A(0) =3500 A. The dashed line is the empirical fitting for pure DPPH, AB~ T -~/2 (see text for discussion ). Inset: temperature behavior of the resonance field value, Br, for ceramic YBa2CusO~. The solid line represents a guide to the eye. The obtained Tc value is 93 K.

on the sample surface over a wide range of temperatures. It should be noted that there were no "'steps" on the AB(T) curve below T=240 K for DPPH radicals deposited on ceramic surface in contrast to the results of ref. [23 ]. The linewidth value just before the superconducting transition is 2.0 G, a factor of 5 smaller than the linewidth reported in ref. [23]. It is possible that the above mentioned step-like broad.... L~Aa . . . . ,.~ ~.o ,,. 1o.,;..,, ~ , ¢ 1 1 1 1 1 ~ ko, IkJtl~IIIJWY -"II'IJ v I'ql~ I~¢t,,IIUlII~I, I[,Jtl~¢ ~.,,.;~,,.,o.~ LI, IlIKIII~Lt~U LIIOt ,ho LIII~¢ intrin sic properties of the sample prepared. Two methods of DPPH deposition on a ceramic surface were used. A concentrated solution of a solid DPPH radical in acetone was poured on the powder of high-To ceramics. Moistened samples were kept in the open air until the acetone evaporated. The other preparation method was based on grinding high-T¢

Yu.N. Shvachko et al. / Detection of the flux-line lattice by ESR probe

ceramics together with a small amount of organic radicals. In order to avoid the vibration of superconducting particles in the sample tube, the samples were fixed in paraffin. A stronger signal was obtained for samples prepared by the latter method, while the former method was expected to give a closer contact between DPPH and the surface of the ceramics. The results of measurements of the ESR linewidth and the resonance field in YBa2Cu307 and Tl2Ba2Ca2Cu3Olo are presented in figs. l and 2 and in the respective insets. Some of the depicted experimental points just below Tc correspond to samples prepared by using the first method. The data for lower temperatures correspond to samples prepared by powder grinding, because of the more intensive signal for these samples. Measurements of the integrated intensity for both

24 0

0

0

0

2O

,,•16 °w=l

tO0

200

T, K

~

300

8

449

high-Tc compounds investigated show a Curie behavior down to the Tc value, then a plateau region within about A T = 4 0 K and, again, a Curie-like behavior with a different Curie constant at low temperatures. We feel that such behavior reflects the disappearance of a certain amount of moments inside the superconducting volume and that such a dependence may be used to monitor the amount of radical that takes part in ESR signal formation. The sharp increase in ESR linewidth, observed both for YBa2Cu307 samples below T¢=93 K and Tl2Ba2Ca2Cu3Oto below T¢= 122 K, may obviously be attributed to the appearance of FLs within the bulk of the superconductor. The appearance of a monotonous line shift below these temperatures also confirms the assumption made. It is reasonable to suppose that there are no other mechanisms of signal broadening than the inhomogeneous magnetic fields of FLs which influence the ESR line at 20 K < T< T~. However, at T< 20 K the line broadening governed by the intrinsic DPPH mechanisms (see AB(T) for pure DPPH) arises and the additional low temperature segment in the AB(T) curves (figs. 1 and 2) may be regarded as a combination of both mechanisms of broadening. For both compounds, we have not observed any other ESR signals that can bc attributed to the nonsuperconducting phases [27] except a narrow DPPH line. Besides the resonance signal, a hysteresis nonresonant microwave absorption was also observed. The value of the hysteresis loop was of the same order of magnitude as the ESR signal amplitude, and these phenomena have been used for superconductivity monitoring. Details of the hysteresis phenomena can be found elsewhere [24-26].

Q I:1

0

=

0

I

tO0

I

I

ra

i

200

4. Discussion I

300

Temperature, K Fig. 2. Temperature behavior of the ESR linewidth in ceramic Tl2Ba2Ca2Cu30~o. The solid line refers to fitting results with ,;.(0)=3900 A at, d To= 122 K, which also include signal broadening in pure DPPH at low temperatures. Inset: temperature behavior of the resonance field, B, in Tl2Ba2Ca2Cu30~o ceramics. The solid line is a guide to the eye. The estimated Tc value is 88 K, the position o f t h e Br(T) minimum is at 35 K.

The absence of strong spin-spin correlations in the DPPH system allows one to consider the picture of non-interacting local magnetic moments put in an inhomogeneous magnetic field, a picture similar to the NMR case. It was Pincus et al. [ 14 ] who first suggested treating the FLL contribution to the NMR signal in terms of the second moment of the gaussian line:

450

Yu.N. Shvachko et ai. / Detection of the flux-line lattice by ESR probe

2 -t/2

¢o it2(16n3) I/2,

( 1)

where d is the FLL parameter within the square-lattice approximation,/~ the external magnetic field and 0o=2.068X 10 -7 G cm 2 the flux quantum. The flux-line contribution is an additional source of ESR line broadening and can be simply extracted from the expression for the observed ESR linewidth: AB~= (AB2 + ~ 2 )

'/2 ,

(2)

where AB, is the observed linewidth, and ABo the linewidth due to other relaxation mechanisms. We believe that, except eq. ( 1 ), there are no other sources that broaden ~he DPPH ESR line on the sample surface. So ABo(T) is the linewidth for pure DPPH powder (see fig. 1 ). A more detailed description has been made on the basis of Brandt's approach [20 ]. It was shown that for impure superconductors with it>> ~, a simple London picture independent of superconductivity mechanisms is valid [21 ]. Therefore, the value ( ~ 2 ) might be calculated (for the triangular FLL) as: (AB2) =0.00371022 -4

for

b=B/Bc2 ~<0.25, (3)

(~--B2) = 7 . 5 2 × 10-4( 1 - b ) 2 x [ 1+ 3.9( l -b)2l¢~it-4

for b >i 0.7.

(4)

The magnetic field needed to record the ESR signal does not exceed 3.4 kG, and for temperatures t=T/T<~0.9 the condition b<0.25 is fulfilled well and eqs. (2) and (3) can be used for data fitting. In order to analyze the obselved AB(T) behavior more properly, we suggest a qualitative model. This aporoach analogous to that developed for ~t+SR experiments [21] has been undertaken step by step within a cruder approximation. Perfect square FLL was considered in order to obtain the surface distribution of local magnetic fields, Bs(r): B~(r)= Y" B , j ( I r - r , j [ ) , t,I

i , j = l .... ,7

(5)

where B,j(Ir-rol ) denotes the field contribution from the vortex located in the r,j= (a,, bj) position. In order to make the present model more lucid, the solution of the Ginsburg-Landau equation was ignored, especially because then the London limit appears to be applicable under the experimental conditions discussed [ 21,28 ]. The approximation for the single FL, which does not involve a vortex core, was used:

Bo( Ir-r~jl ) =B* .i r_ ro l

,

exp

-

I r- r° l it

.

(6)

Here the vector r o determines the FL-center. The modified core field value, B*, does not reflect the saturation of the Bo-rise which results from the GLsolution for r-, r u and Ir - rol ~ l~. Nevertheless, it allows one to reach a match between eq. (6) and the exact solution of the GL equation for a certain penetration depth 2. Thus the quantity B* has become another parameter that is needed to get a reasonably good fitting for the range I r - rol > ~. Note that both corrections mentioned before can be considered strictly. However, this does not improve the agreement with the predicted B(r). The obtained two-dimensional distributions, Bs(r), for example presented in the inset of fig. 3 for certain experimental conditions (d=2000 A, it=3000 A) demonstrate pictures similar to exact ones extracted from the GL equation [ 22 ]. The density of local fields Dt (B) was calculated by counting the points with equal values Bs(r) over the unit cell of FLL. The corresponding n t (B) curve is depicted in fig. 3. It is seen that the contribution of high value magnetic fields/l~ B* does not play a significant role in the ESR line broadening, because it forms the right wing of the distribution nt(B) rather far from its maximum. Therefore the above discussed discrepancy for the single vortex approximation within the range Ar.<.~ (eq° ( 6 ) ) seems to be unimportant. The calculated average quantity (B>, which corresponds to the external field/1 without diamagnetic shielding effects, appears to be larger than the field of the nt(B) maximum: (Bma~- ( B > ) / B * = 0 . 0 1 5 . Therefore, the expected shift of the n t (B) maximum will bring the ESR line to higher magnetic fields, thus

Yu.N. Shvachko et al. / Detection of the.flux.line lattice by ESR probe

Such a convolution suggests the superposition of the external magnetic field,/~, and FLL fields n t(B' ) at the position where the paramagnetic centers lie. The external field sweep provides conditions for an ESR observation, while the FLL fields are viewed as a static distribution. It is reasonable to suppose that those centers which do not feel the external field will give a non-resonance response rather than a resonance signal. The calculated ESR line shape is shown in fig. 4 where the dimensionless value ( / ~ - < B > )/B* corresponds to the x-scale. One can see that the obtained line asymmetry, 1.7, does not exceed 2 for different ;ts and ds even for a perfect square FLL. This means that the asymmetric slopes of the n t(B) distribution and the low-field cut-offas well mainly form the wings of the ESR signal which are close to the noise level. The experimentally observed asymmetry

12-

<

45 !

8

m

V

4

0.0

2o[.

O.1

(Bs-(B))/B" Fig. 3. Field densities for a perfect square flux-line lattice (FLL) (d=2000 ~,, A=3000 ~,). Curve 1 (nn(B)): perturbed by strong (Sr/d=O.02) pinning. Curve 2: (n2(B)). Curve 3 represents the field density, n3(B), at the point shifted out of the superconducting surface (see text for discussion ). Inset: lines of constant field foran ideal square FLL (d= 2000 A, ).=3000 A) with the singlevortex approximation obtained from eq. ( 6 ).

decreasing A(T). On the contrary, diamagnetic shielding will cause a negative shift proportional to the diamagnetic response of the superconductor. As a result of their competition, the position of' the ESR line might be both monotonic with decreasing temperature, as was observed for YBa2Cu307 (see the inset in fig. 1) and non-monotonic, as with Tl2Ba2Ca2CusOm (see the inset in fig. 2). These qualitative conclusions are also consistent with the temperature behavior of the ESR resonance field in superconducting alloys below Tc [ 29 ]. The expected ESR line shape was calculated as the convolution of the n i (B) function and of the original Lorentzian line F(B) of width ABo, corresponding to a pure DPPH (see fig. l ): B*

Fl(/~)= t dB'F(B-B')n~(B'). I#

0

(7)

I

) "~ - t o

t

-200.1

1

L

0.0

I

0.1 °

Fig. 4. Calculated ESR line shapes corresponding to field densities depicted in fig. 3. Lines 1,2 and 3 correspond to n~(B), n,(B) and n3( B ), respectively. The asymmetry ratio for the original signal (curve 1 ) is 1.7, while for distorted FLL (pinning) it is 1.0. The intensities of the signals presented are equal.

452

Yu.N. Shvachko et al. / Detection of :heflux-line lattice by ESR probe

ratios were 1.1 for YBa2Cu307 and 1.0 for Tl2Ba2Ca2Cu~Olo, values that are substantially less than the calculated ones. One can see that the peak to peak ESR linewidth is formed by an arbitrary short range of fields near the maximum of n (B). Meanwhile calculations predict a strong dependence of the n (B) peak shape on the kind of FLL. The triangular lattice gives a narrower n ( B ) peak than do square or any distorted FLLs [22,30]. That is why the FLL irregularities caused by the pinning effect must be taken into consideration. Thus two features should be pointed out: first, the ESR line below Tc seems to be shifted with asymmetric wings; second, not only 2 and B* parameters determine the ESR linewidth, ABs, but the kind of vortex structure also determines the width of n ~(B), i.e. the ABs value. The influence of strong surface pinning was accounted for by putting into consideration the random displacements, 6r, of the vortex centers around their position of equilibrium [22,30l. Figure 3 (curve 2) demonstr~.tes the distribution n2(B) calculated for a 20 percent deviation (6r/d=0.2) of FLs in a perfect FLL. The corresponding ESR line demonstrates a more symmetric and a broader shape for the same parameters 2 and B* as those used for n~ (B) calculations. Besides pinning effects, one more important consequence of the experimental conditions should be discussed. Owing to the finite sizes of DPPH grains on the ceramic particles, there always should be magnetic moments at different distances from the superconducting volume. Therefore, some centers will feel a def~.rmed distribution, n3(B), which comes from the changed Bs(r) function. For simplicity, we suggest an exponential evolution of Bs(r) along the (perpendicular to the surface) Z-axis up to the homogeneous value B~(r)--,0 when Z-~ ov (B~= 0 reflects the vanishing of the local fields at the distant points). The obtained distribution, n3(B), and the ESR line presented in figs. 3 and 4 for Z = x (where r is a constant decrement), demonstrate a tendency of the reverse pinning effects: the ESR line has become narrower and more symmetric. However, the maximum of n3 (B) continues to stay at a lower position than the average field, B'a~/ (B)
It is important to point out that according to the above discussion, there are at least two sources which make us modify the ~.(T) estimates extracted from ESR data. On the one hand, the irregularities of the FLL that are caused by pinning effects lead to spreading of the n (B) distribution and, accordingly, to an additional ESR line broadening. In this case the AB(T) dependence treated without pinning effects will give ~. values that are smaller than the true ones. On the other hand, finite sizes of DPPH grains will mask the inhomogeneous broadening of the ESR signal, and the corresponding ~. value will exceed the true estimates. We believe that both of the aforementioned difficulties have come from the quality of the sample and the quality of decoration. In addition to the features discussed, one should note that, owing to the layered structure of the highTc compounds investigated, the typical granule has a plate-like shape. So it is reasonable to suppose that a greater portion of the DPPH moments appears to be deposited on the a-b plane of the micro-crystals. Therefore, the obtained ~[estimates reflect an incomplete average of 2 values over different orientations, in contrast to bulk-sensitive techniques such as ~t+SR. In figs. 5 and 6 we present the comparison of experimental data treated by eqs. ( 1 ) and (2) with the theoretically predicted dependence for the BCS model [28 ]: 2( T ) - ' = 2 ( 0 ) - ' ( 1 - (T/Tc) 4) ,/2.

(8)

The observed 2(T) dependence for YBa2Cu307 ceramics shows a good coincidence with the theoretical curve over a wide temperature range. It was also found that in spite of the accuracy in the determination of 2 (0) (62 = 100 A ), the estimated 2 values for different samples were largely different. We believe that these deviations might be attributed to the quality of the samples, and the measured ~ (0) value for YBaECU307is 3500 A. It is a higher estit~aate than those obtained from other techniques. One possible explanation of this divergence is the existence of a non-superconducting layer on the surface of the granules or the influence of the finite sizes of DPPH grains. A similar but slightly more complicated behavior of 2(T) has been obtained for ceramics Tl2Ba2Ca2Cu30~o. A comparison of fig. 2 and its inset indicates that the Tc values measured by the re-

Yu.N. Shvachko et ai. / Detection of the flux-line lattice by ESR probe

453

rl

ra~~ t~tl o ~ n 13

~

o "!

2

\

!

!

!

~<

,<

00

i

,

I

I

50

I

100

Temperature, K

O0

l

I 50

i

l tO0

Temperature, K

Fig. 5. Change in the mag::etic penetration depth extracted from fig. 1 by using eq. ( ! ). The solid line represents the result of fitting on the basis ofeq. (8).

Fig. 6. Change in the magnetic penetration depth obtained from fig. 2 using eq. ( 1 ). The solid line has been fitted on the basis of eq. (8).

sistivity technique, ESR resonance field and linewidth appear to be different. Another reason for the not so good 2(T) fitting is the possible presence of superconducting phases with different To= 80,110,120 K; this is known to take place in TIbased compounds, particularly near the surface of the samples investigated. We feel that for this reason the linewidth behavior AB(T) within the intermediate temperature region shows a quasi-linear increase down to helium temperatures (see fig. 2 ). The best fit for Tl2Ba2Ca2CusO~o was obtained for To= 122 K (resistivity data), the corresponding (0) value being 3900 A. The additional broadening below 22 K was well described by the internal relaxation in DPPH (see fig. 1 ). The comparison presented in fig. 6 indicates that in spite of the possible multiphase nature of Tl-based samples, the observed 2(T) behavior seems to be close to the prediction of BCS.

5. Concluding remarks The work reported provides measurements of magnetic penetration depth in Y B a 2 C u 3 0 7 and Ti2Ba2Ca2Cu30~o high-To superconductors on the basis of the ESR-probe decoration technique. It is shown that the linewidth broadening and the resonance field shift below Tc extracted from ESR experiments can be treated by using the local field det,,sity approach developed for ~t+SR and NMR techniques. We consider a qualitative model of the FLL influence on the ESR spectrum formation, Calculated 2 values are 3500 A in YBa2Cu307 and 3900 A in T!2Ba2Ca2Cu~Olo. The difference between our estimate of 2 in ceramic YBa:CusO7 and that from other techniques could be due to the incomplete averaging over the 2s because of the non-uniformly distributed organic radical over the surface of the superconducting granules. Temperature dependences of the magnetic pene-

454

Yu.N. Shvachko et al. / Detectwn of the flux-line lattice by F~SRprobe

tration depth in both compounds investigated demonstrate a behavior close to ( l - ( T / T c ) 4 ) - ~ / 2 in agreement with the predictions of BCS. We believe that the technique considered will be a useful tool for the investigation of FLs distribution in both single crystals and superconducting films.

Acknowledgements We wish to thank Prof. V.F. Gantmakher for useful comments and Dr. V.M. Laptev for helpful discussions.

References [ i ] F.N. Gygax, B. Hitti, E. Lippelt, A. Schenck, D. Cattani, J. (?ors, M. Decroux, O. Fischer and S. Barth, Europhys. Lett. 4"(1987) 473. [2]G. Aeppli, R.J. Cava, E.J. Ansaldo, J.H. Brewer, S.R. Kreitzman, G.M. Luke, D.R. Noakes and R.F. Kiefl, Phys. Rev. B 35 (1987) 7129. [3] W.J. Kossler, J.R. Kempton, X.H. Yu, H.E. Schone, Y.J. Uemura, A.R. Moodenbaugh, M. Suenaga and C.E. Stronach, Phys. Rev. B 35 ( ! 987) 7133. [4] D.R. Harschman, G. Aeppli, E.J. Ansaldo, B. Batiogg, J.N. Brewer, J.F. Carolan, R.J. Cava, M. Celio, A.C.D. Chaklader, W.N. Kempton, S.R. Kreitzrlan, G.M. Luke, D.R. Noakes and M. Senba, Phys. Rev. B :.~6(1987) 2386. [5]W.J. Gallagher, T.K. Wcxthington, T.R. Dinger, F. Holtzberg, D.L. Kaiser and R.L. Sandstrom, Physica B 148 (1987) 228; R. Felicci et al., preprint Rutherford-Appleton Lab. ( 1987 ) RAL-87-062. [6] P. Monod, B. Dubois and P. Odier, Physica C 153-155 ( i 988 ) 1489. [ 7 ] J.R. Cooper, M. Petravic, D. Drobac, B. Korin-Hamzic, N. Brnicevic, M. Paljevic and G. Collin, Physica C 153-155 (1988) 1491. [ 8 ] P. Peyrals, J. Rosenblatt, A. Rabouton, C. Lebeau, C. Perrin, O. Pena, A. Perrin and M. Sergent, Physica C 153-155 ( 1988 ) 1493. [9] A. Schenk, Physica C 153-155 (1988) 1127.

[101 D.R. Harshman, L.F. Schneemeyer, J.V. Waszczak, G. Aeppli, R.J. Cava, B. Batlogg, L.W. Rupp, E.J. Ansaimo, D.LI. Williams, Phys. Rev. B 39 (1989) 851. [111 L. Krusin-Elbaum, R.L Greene, F. Holtzberg, A.P. Malozemoff, Y. Yeshurun, Phys. Rev. Lett. 62 (1989) 217; E.M. Jackson, S.B. Liao, J. Silvis, A.H. Swihart, S.M. Bhagat, R. Crittenden, R.E. Glover and M.A. Manheimer, Physica C 152 (1988) 125; A. Umezawa, G.W. Crabtree, J.Z. Liu, T.J. Moran, S.K. Malik, L.H. Nunes, W.L. Kwok and C.H. Sowers, Phys. Rev. B 38 (1988) 2843. [121 P. Birrer, F.N. Gygax, B. Hettich, B. Hitti, E. Lippelt, H. Maletta, A. Schenck and M. Weber, Physica C 158 (1989) 230. [13] M. Mehring, F. Hentsch, Hj. Mattausch and A. Simon, Z. Phys. B. Cond. Mat. 77 (1989) 355. [141 P. Pincus, A.C. Gossard, V. Jaccarino and J.H. Vernick, Phys. Lett. ! 3 ( ! 964) 851. [151 J.M. Delrieu et al, J. Phys. F. 3 ( i 973 ) 893. [16] A.G. Redfield, Phys. Rev. 162 (1967) 367. [171 P. Thorel, R. Kahn, Y. Simon and D. Criber, J. Phys. 34 ( 1973 ) 447. [181 J.R. Clem, J. Low Temp. Phys. 18 (1975) 427. [191 J.R. Cooper, C.T. Chu, L.W. Zhou, B. Dunn and G. Gruner, Phys. Rev. B 37 (1988) 638; D.W. Cooke, R.L. Hutson, R.S. Kwok, M. Maer, H. Rempp, M.E. Schillad, J.L. Smith, J.O. Willis, R.L. Licht, K.-C. Chan, C. Boekema, S.P. Weathersby, J.A. Flint and J. Oostens, Phys. Rev. B 37 (1988) 9401. [201 E.H. Brandt, Phys. Rev. B 37 (1988) 2349. [211 E.H. Brandt and A. Seeger, Adv. Phys. 35 ( 1986 ) 189. [221 E.H. Brandt, J. Low Temp. Phys. 73 (1988) 355. [231 B. Rakvin, M. Pozek and A. Dulcic, Solid State Commun. 72 (1989) 199. [24] A.M. Portis, Int. School of Mater. Sci. and Techn., 16th course, eds. J.G. Bednorz and K.A. Miiller (Springer, Heidelberg, 1989). [25] Yu.N. Shvachko, D.Z. Khusainov, A.A. Romanyukha and V.V. Ustinov, Solid State Commun. 69 (1989) 611. [26] A. Dulcic, B. Leontic, M. Peric and B. Rakvin, Europhys. Lett. 4 (1987) 1403; M. Peric, B. Rakvin, M. Prester, N. Brnicevic and A. Dulcic, Phys. Rev. B 37 (1988) 522. [27]A.A. Romanyukha, Yu.N. Shvachko, V.Yu. Irkhin, M.I. Katsnelson, A.A. Koshta and V.V. Ustinov, Physica C ! 71 (1990) 276. [28] A.A. Abrikosov, Fundamentals of the Theory of Metals (Elsevier, Amsterdam, 1988 ) 630 pp. [29] S.E. Barnes, Adv. Phys. 30 ( 1981 ) 801. [30] P.H. Kes, Physica C 153-155 (1988) 1121.