Low-frequency AC electrical resistivity of granular Y1Ba2Cu3O7−δ compounds

Physica C 176 ( 199 I ) 49-56 North-Holland

Low-frequency AC electrical resistivity of granular Y 1Ba2Cu307 _ 6 compounds F. MiguClez, J. Maza and FClix Vidal Laboratorio de Fisica de Materiales, Santiago de Compostela. Spain Received Revised

Departamento

13 November 1990 manuscript received 5 February

de Fisica

de la Materia

Condensada,

Universldad

de Santiago,

15700

199 I

The behaviour (amplitude and phase) of the AC electrical resistivity, p(u) of five bulk polycrystalline Y IBa2Cu307--6 single phase (to within 4%) samples with 610.1 and T,,> 90 K, but with very different granular structure, was studied quantitatively up to 120 kHz. The measurements were performed in the temperature range of room temperature down to below the superconducting transition, and at low effective current densities (up to 25 A/cm2) and applied magnetic fields (up to 30 mT). In all cases, within a resistivity resolution of 1 pR cm, no frequency-dependent effects were observed. Appreciable localization effects, including hopping conductivity, must therefore be excluded in the a&plane of YiBazCu,O,-a materials, for the low energy range analysed.

1. Introduction Knowledge of the AC behaviour of the electrical resistivity, p, at “acoustical” frequencies (v < 120 kHz), of the copper oxide superconductors is important for fundamental as well as for practical reasons. The frequency dependence of the amplitude and of the phase of p( u), in both the superconducting and normal state, may be used to probe various possible intrinsic effects as, for instance, those associated with the carriers’ effective mass [ 1,2 1, or with various localization effects, including hopping conductivity [ 3 1. Although this type of microscopic effects on p ( v ) are expected to be appreciable only in the ultrasonic or hypersonic frequency ranges (Y> 1 MHz), results in some “precursor” (which does not become superconducting at any studied temperature) copper oxide materials (for instance, La2Cu04+y or YBazCu306) suggest that they cannot be excluded in “high temperature” copper oxide superconductors ( HTSC ), even in the low frequency range [ 4-7 1. In addition, p( v) in HTSC may also be affected by various kinds of macroscopic electromagnetic effects, as hysteresis or eddy-current losses, as is the case in “low temperature” superconductors [ 8 1. 092 l-4534/9

I /$03.50

0 199 1 - Elsevier

Science

Publishers

In addition to these possible intrinsic and very general frequency-dependent effects (which will be also dependent on the layered nature of these copper oxide materials), p( v) in HTSC may also be affected in various other specific but important situations. We have, for instance, the case of the presence of an external magnetic field, which may introduce new length and energy scales,in particular associated with the vortex lines, that may be relevant to p(v) at acoustical frequencies. For example, as in the case of inhomogeneous low temperature type II superconductors [ 9 1, the vortex pinning may depend on the electrical current frequency. The presence of weak links, in the case of granular samples, or of critical phenomena near the normal-superconducting transition, may also produce new frequency-dependent effects on the AC resistivity. For instance, hoppinglike effects may appear, even at low frequency, in the intergrain barriers. To the best of our knowledge, the first quantitative data on p(v) in bulk copper oxide materials having a superconducting transition were those published by Veira et al. [ 10,111 for granular LnBazCu30,-6 (LBCO) compounds, with Ln=Y or Ho. These measurements were performed in zero applied mag-

B.V. (North-Holland)

50

F. Migudez et al. /Low-frequency AC resistivity of Y,Ba2Cu30,-d

netic field and in the very low frequency range, 0 < v < 2 kHz. To within the experimental resolution (&= k 2 $2 cm), the absolute resistivity was found to be frequency independent over the entire temperature range (TI 300 K) and for electrical current densities j&,, I 10 A/cm*, where j;,,,,, is the “effective” current density (see later). Early information on p(v) above T, in granular copper oxide superconductors in the low-frequency range (v < 30 kHz ) has also been published by Schlesinger et al. [ 12,13 1, but their results were very qualitative and somewhat ambiguous: whereas in their first paper [ 121 an appreciable frequency dependence was noticed, in a second paper [ 13 ] these effects were attributed to the poor quality of the former samples. Besides the fact that an explanation needs to be worked out, in both papers no quantitative information, not even the resistivity resolution, is given. A statement about the absence of appreciable frequency-dependent effects on the AC resistivity below 0.5 GHz in La,$ro.iCu04, which exhibits bulk superconductivity below 30 K, was also published by Maglione et al. [ 41. More recent measurements focus only on the qualitative aspects of the amplitude of p(v) in YBaCuO samples, also in the absence of an applied magnetic field [ 14,15 1. These qualitative data confirm the first results of Veira et al. lo] and seem to extend up to 1 GHz the frequencies for which no appreciable frequency-dependent effects on the absolute resistivity are observed in full oxigenated YBa2Cu307--6 samples. However, Behrooz and Zettl [ 141 seem to observe appreciable differences between PAo and pbc near and below the superconducting transition, although they attribute these differences to spurious effects. All the above-mentioned AC measurements concern bulk single-crystal or polycrystal samples having a superconducting phase. Some results on the complex AC conductivity in the hypersonic frequency range (60 GHz) .are also available for HTSC films [ 161. These data indicate that in the region well above T,, the transmission loss follows the observed trend of the DC resistivity. However, the AC and DC behaviours are very different around the superconducting transition. Also, there is no phase-shift well above T,, but at lower temperatures a large negative phase shift was observed. In this paper we will present quantitative data of both the amplitude and the phase of p(v) up to

v= 120 kHz in bulk polycrystalline Y,Ba2Cu307-6 single phase (to within 4%) samples with 610.1 and with T,,>90 K, but with very different granular structure. The measurements were performed in the temperature range of room temperature down to below the superconducting transition and at low electrical current densities (up to 25 A/cm’). In particular, we examine in detail two different aspects: the behaviour of the amplitude and the phase of p(v) around the superconducting transition, and the possible influence of granularity. Also, we will present here the first data onp( v), in YBaCuO samples having a superconducting transition and in the very lowfrequency range (up to 0.5 KHz), in the presence of a small applied magnetic field (up to 30 mT) normal to the applied electrical current. 2. Experimental details

As mentioned above, five different single-phase and polycrystalline samples having very different granularity characteristics, and the same nominal composition, YBazCu30,-8, with 610.1, were used. The preparation of these samples and details of their structural, stoichiometric and magnetic properties have been reported elsewhere [ 17 1. The characteristic size of the crystallites ranges from 1 to 10 pm, although in one of the samples a few crystallites of the order of 1OW3mm3 can be observed. Under SEM and optical microscopy all samples studied in this work show pores on the same scale of the grains, the latter showing also a high density of twin boundaries at length scales larger than 1000 A. The porosity of the samples, directly observable in some cases by simple visual inspection, decreases the average density to SO-95% of the ideal one. The length scales of these different structural inhomogeneities are, in all cases, bigger than other characteristic lengths in the system relevant for p, such as the grain average meanfree path of the normal carriers, or the superconducting correlation length amplitude in all directions. In the opposite side, the electromagnetic skin depth in our samples is, for w/2x1 120 kHz, of the order of 10 cm or larger i.e. much larger than any of the characteristics lengths indicated above, and even larger than the typical sample length (1 cm). In the DC case, we have found that the influence of these long-scale structural inhomogeneities, and of the in-

51

F. Migudez et al. /Low-frequency AC resistrvity of YIBa2CuJ07-6

tergrain coupling, on the resistivity may be taken into account by using a simple phenomenological picture, in terms of an effective cross section of the sample (which takes also into account the conductivity-path lengthening, see below), and an overall intergrain resistance [ 18-201. One of the aims of the present work is, in fact, to check quantitatively the validity of that picture in the low-frequency AC case. In table 1, we summarise some of the most relevant characteristic of our samples. These data were obtained from DC resistivity measurements, and the notation is the same as in refs. [ 191 and [20]. In particular, dp/dT is the slope of p( T) between 150 K and 250 K, a temperature range were p(T) may be fairly well approximated by a straight line. T, is defined by p( T,) ~0, 7’,, is the temperature where p(T) around the transition has its inflexion point, and AT,, is the upper half-width of the resistive transition [ 19 1. Note that whereas T, will be appreciably affected by the granular nature of our samples, T,, is expected to be close to the mean-field-like normalsuperconducting transition temperature of the grains [ 19,201. So, the differences in T,, for the various samples in table 1 are probably due to small differences in their oxygen content. Much more important are the differences in their normal DC resistivity, poc( 300 K), or in the temperature slope of pDc( T) in the normal region far away from T,,. As indicated before, these differences may be easily explained in terms of a phenomenological picture, similar to that first proposed by Kirtley and coworkers for granular LTSC [ 18 1, that takes into account the granular nature of our samples [ 19,201: The measured DC resistivity above Tcr,and in zero applied magnetic field, is related to the intrinsic resistivity~f~ in the &plane of an ideal single crystal by

P(T)=

;P~&-)+P,,

.

(1)

The fact that p(T) depends only on phh is because the resistivity in the c direction is orders of magnitude larger than in the &plane [21 1. In eq. (1 ), p (0

+ciT,

(2)

or by the temperature dependence proposed by Anderson and Zou [ 23 1, p&,(T)=?,/T+C;T.

(3)

Table 1 Summary of the most relevant characteristics of the samples as deduced from DC electrical measurements Sample A B C D ‘E

~(300 K) (mRcm)

QldT W cm/K)

TC WI

TCI (K)

AT,, (K)

p(Tc,)

35.5 11.5 3.3 1.5 0.67

92 31 8 4 2

89.50 90.90 88.40 90.60 90.20

91.05 91.45 90.55 91.20 90.85

0.35 0.20 0.35 0.25 0.25

7.65 1.76 0.94 0.27 0.10

.0x lo*

(mRcm)

PCI

(mQcm) 0.59 1.65 6.40 11.62 27.08

9. I 2.24 0.98 0.24 0.10

52

F. Migudez

et al. /Low-frequency

AC resistivity

tached to the samples with silver paste (DuPont 4929 ), and they were mounted so as to minimise inductive pickup. The measurements of the in-phase and out-of-phase AC resistivity relative to the applied AC current were made by using a conventional lock-in amplifier phase-sensitive technique. The current was supplied by the reference output signal of an EG&G Princeton Applied Research model 5210 phase-sensitive detector, which measures simultaneously the longitudinal in-phase (x) and out-ofphase (y) voltage with the transport current. Sometimes, this reference output signal was amplified by a Hewlett Packard amplifier model 6827 A (up to 15 kHz ). By using an FFT Spectrum Analyser Solartron Schlumberger 1201 we have checked that our electronic system does not introduce any appreciable distortion in the AC current, j,,(y), through the sample. However, the phase of j,, may be appreciably different from that of the initial reference signal (in particular, due to the use of the current amplifier). The phase of j,o through the sample is, therefore, determined by using as phase-calibration signal the direct inductive pickup voltage appearing on a coil placed near one of the current leads of the sample. Such a calibration signal must present a phase shift of lc/2 rad with the AC current in the sample. The relative AC voltage, SL’, and phase resolutions were better than 0.1 uV and lo, respectively. Cor-

In both cases, the rms deviation is less than 1% for all the different samples studied. Indeed, the use of eqs. (2) or (3) change c, a lot (and also the normalconductivity physics) but c2 only slightly. But the important point here is that, as was stressed in fig. l(a) of ref. [ 191, when eqs. (2) or (3) are used in eq. ( 1) to extract p and pet in polycrystals, one obtains almost the same values for these parameters. Although we do not analyse in this work the paraconductivity, Ao( T) above Tcr, let us just note that the use of eqs. (2 ) or (3 ) to extrapolate the background conductivity to T,, does not have an appreciable influence on Ao( T) either, as was also recalled in refs. [ 191 and [ 201. However, as the most recent data seem to suggest a slightly better adequacy of the linear behaviour, in the present work we use eq. (2 ) with c\ =5 pRcm and ci ~0.5 pQcmK-‘, which corresponds to the average values from the data of refs. [21], [22] and [24] (Ci,=(0.5~0.2) uQcmK-’ and cl =(5~ 10) l&cm). Electrical transport in (random oriented) granular samples will thus probe the intrinsic conductivity in the &plane, and also the non-intrinsic parameters p and pc,. For resistivity measurements, rectangular parallelepiped-shaped samples were cut out from the original disc-shaped samples. The typical dimensions were 1 x 1 X 15 mm3 (see inset in fig. 1). The alternating-current and voltage coaxial leads were at-

V out-of V

in-phase dc

of Y,Ba2Cu307-d

-phase ; 100 KHz ; 100 KHZ

0.oe

0.

-6

0.0* 0G*

-4 -2

-0 0

100

200

300

T (K) Fig. 1. An example (sample the phase of the AC current.

C in table 1) of the in-phase (open circles) and out-of-phase (open The DC voltage (solid circles) agrees with the in-phase voltage.

squares)

voltage

signals

with respect

to

F. Migudez et al. /Low-frequency AC resistivity of Y,BaZCujO,-s

respondingly, as in our experiments the distance, d,, between the voltage leads were typically of 1 cm, AC resistivity resolutions were of the order of 1uR cm for the amplitude and 1’ for the phase relative to the applied AC current through the sample. From &a= -Sp/p*, we see that any frequency-dependent effect on verifying the conductivity, Aa=cr(~) -cr(O)
(4)

i.e. the experimental &a(v) resolution is, for a given SV, j and d,, proportional to p2. In the case of YBa2Cu@-s samples, where for instance, at T= 300 K, p:,, 20.2 mQcm, and using also j=25 A/cm*, d, = 1 cm, and 6 V= 0.1 uV, it will be possible to resolve 60;~ = 1O- ’ (R cm ) - ‘. The DC resistivity has been measured by using a 84 digits Solartron 7081 nanovoltmeter and the resolution was 1 uQ cm. The temperature resolution was of the order of 10 mK.

3. Experimental results and discussion A typical example of the in-phase and out-of-phase voltage signals with respect to the phase of the AC current through the sample is shown in fig. 1. This example corresponds to sample C in table 1, in the absence of an applied magnetic field, and the rms applied current density was 7.6 x 1Om4A/cm*. As for this sample pz7.3~ lo-*, the effecctivecurrent density, defined by

I jc= 2ps, was 0.5 A/cm*. In eq. (5), I is the total applied current, S is the sample section, and the factor 2 takes into account that, as noted before, p in eq. ( 1) is associated not only with a reduction of the effective cross section of sample but also with the random orientation of the a&planes of the different grains [ 19 ] _ For comparison, we also show in fig. 1 the DC voltage corresponding to an applied DC current density of 7.6x 10P4 A/cm*. In fig. 2, the DC and th (absolute) AC resistivities (for v= 100 kHz) of the five samples of table 1 are compared. The AC and DC effective current densities, j’, used in all measure-

53

ments were well below the values for which p(T) above T,, shows a je dependence. The results of fig. 2 show that, to within the experimental resolution indicated in section 2, pAC and pDC are the same. An additional check of absence of appreciable frequency-dependent effects in the AC normal resistivity is presented in fig. 3. In this figure, the (absolute) AC resistivity at T=300 K of three of our samples is plotted as a function of the frequency v of the AC applied current density. These date were obtained with je = 20 A/cm*. Here again no AC effects are observed in all the frequency range studied (up to 120 kHz). From the data of figs. 1 to 3, we may, therefore, conclude that above the superconducting transition of the granular YBa2Cu,0,-J samples studied: ( 1) The amplitude of the AC resistivity is the same as the DC resistivity. (2) No phase shift is observed. These conclusions apply to the ranges: j:,, 525 A/cm’; TI 300 K; VI 120 kHz. When analysing these results in terms of eq. ( 1). we may conclude that the absence of measurable effects concern the intrinsic resistivity in the ah-plane, as well as the intergrain links. The first conclusion confirms quantitatively. in the low-frequency range (up to 120 kHz), the qualitative results of Behrooz and Zettl for YBa2Cu@-,s superconducting single crystals and of Ho et al. for Bi-CaSr-Cu-0 oriented films. Our results for the amplitude and phase of PAC.rule out, in particular, the relevance of any intrinsic frequency-dependent mechanism, such as localization or hopping, in the ahplane of YBa2Cu,0,,- (6s 0.1) materials, for the lowenergy range corresponding to the frequencies inspected. One may wonder how these results in the normal phase of superconducting copper oxide materials compare with those obtained in the nonsunerconducting YBa2Cu30b [ 71 or La2Cu04+,, [ 5 J precursors. As the microscopic mechanisms of the electrical transport in the normal state in these compounds are still not well-established [ 25 1, we just are going to recall here that the frequency-dependent effects observed in precursors, if they were also present with the same absolute strength, are too small to be observed in the much less resistive normal phase of the copper oxide materials having a superconducting transition (see section 2). For instance, the results

54

F.

Miguklezet al. /Low-frequency

5

AC resistivity

I

of Y,Ba2CuJ0,-d

I

8

40

T (K) Fig. 2. DC (open symbols)and 100 kHz AC resistivities (closed symbols) of the five samples of table 1.

0

1

I 2

, 3

( log v

I 4

, 5

I

, 6

(Hz)

Fig. 3. AC resistivity at T= 300 K of samples C, D and E of table I, plotted as a function of frequency, u.

of Chen et al. in La2CuOq+y materials [ 51, also obtained directly from resistivity measurements, indicate that at low temperatures lo-‘(Rem)-‘
terials. As indicated in section 2, such a Ao( v) effect will , if present, probably be only appreciable in the c direction of YBa2Cu307--6 samples, the intrinsic pc being orders of magnitude larger than in the ab-plane [ 2 11. However, for the same reason, we have no access to pc( v) of the polycrystalline samples studied here. Low-frequency AC experiments in copper oxide single crystals are now in progress in our group. An example of our detailed inspection of the AC resistivity around the superconducting transition is shown in fig. 4(a) and (b). This example corresponds to sample E. In fig. 4(b), we show the temperature derivative of the curves of fig. 4(a). The same results have been obtained for all the samples, current densities (up to j&,,, 125 A/cm2) and frequencies studied (up to 120 kHz). Therefore, our result, show quantitatively the absence, to within our resolution, of frequency-dependent effects on the resistive normal-superconducting transition in granular YBazCu307-s samples in the audio-frequency range. Such a conclusion applies to both sides of T,, (a temperature that is supposed to be close to the mean-field critical temperature Tco, see ref. [ 193 ). As a further check of the region around T,,, and most particularly, of the role of the intergrain links in the AC behaviour, we have measured the longitudinal DC and AC electrical field in the presence of a small magnetic field, H, applied normal to the elec-

F. Migudez 3001

I

et al. /Low-frequency

I

I

I

AC resistivity

of YIBa2Cu307_d

!xtr

I . o A

A” 30

/ 91

3 >

0.2

E

0.1

400

I 93

80 KHZ

32

93

T (Kl

Fig. 4. An example (sample superconducting transition: frequencies; (b) Temperature ity in (a).

I

,

87

.

I .A

00’

A. 0.

.” 0

.**

.** : : 100

200

300

z

89

;*

91

’

93

95

Fig. 5. An expanded view around r,, of the longitudinal DC and AC electrical field in the presence of a small magnetic field applied normal to the electrical current. In the inset is shown the Em-phase at u= 37 Hz without magnetic field. This example corresponds to sample C in table 1.

0 91

.

T (Kl

3

30

.*

1

32

0

89

.-

0 0

T (K)

2 2

I

1, = 0: v = 37 HZ H = 30 mT; v = 37 Hz H = 30 mT; v = 325 Hz

: 2a JIIl wc

089

55

E) of the AC resistivity around the (a) In-phase resistivity at different derivative ofthe in-phase resistiv-

trical current. In order to separate the possible effects associated with moving vortex lines (i.e. flux flow) from those due to the intergrain links, the applied magnetic field was always less than 30 mT, i.e. lessthan the average H,, amplitude in YBaCuO samples [26]. So, in principle, for T-=KTcr, no vortices will penetrate into the bulk grains, although, indeed, H,,(T) and the magnetic field affecting the intergrain weak links vanish at the transition. Also, due to the demagnetisation factor we cannot exclude the presence of some small mixing regions at low temperature. In fig. 5, we present an example of our first results. These data correspond to sample C. The important result here is that, for the very low-frequency range examined ( YI 325 Hz), no frequency-dependent effect is observed. Although more measure-

ments as a function of both H and v are obviously needed, it is very probable that the small H-dependent effects observed between T,(H) and T,, are just associated with the breaking of some of the intergranular weak links”’ [27], and not with moving vortex lines, this last mechanism being probably much more sensitive to frequency. For instance, the vortex pinning may depend, as in the case of low temperature type II superconductors [ 91, on the electrical current frequency.

4. Conclusions We have presented quantitative data for both the amplitude and the phase of the AC resistivity, p(v), in bulk granular YBazCu@-8, with 610.1. These measurements cover the ranges 605 TI 300 K; v I 120 kHz; j&,,, -< 25 A/cm*. Some of the measurements were done in the presence of a small applied magnetic field, up to 30 mT, normal to the electrical current. The experimental resolution was 100 nV for the in-phase and out-of-phase longitudinal voltage and 1’ for the relative phases between the voltage and the electrical current. The resistivity resolution was 1 usZcm. The relative temperature resolution was ‘I We observe very similar effects in our samples when increasing the effective current density well above 25 A/cm’, in the absence of an applied magnetic field.

56

F. Migdez

et al. /Low-frequency AC resistivity of YIBa2Cu307--6

10 mK. The measurements were performed in five different samples, all with the same nominal compositions, but having different long-range structural inhomogeneities. In this way, it has been possible to probe the influence of structural inhomogeneities and of the weak links between grains. The central conclusion of this work is the absence of any observable frequency-dependent effect, to within the resolutions and ranges indicated earlier, in granular YBa2Cu307-6 compounds, in both the superconducting and the normal states. These results confirm, in particular, the absence of appreciable (to within 0.1 (Q cm)-’ intrinsic hopping conductivity in the &plane in these compounds at low-energy range [ 14,16 1. But also, our results exclude any important low-frequency electromagnetic effect, even in the temperature region close to the superconducting transition where the p behaviour in granular samples is dominated, on the low-temperature side, by weak links and percolative effects, and by order parameter fluctuations on the high-temperature side. On both sides on the transition, the phase of the measured longitudinal electrical field is the same as the phase of the AC applied electrical current. This rules out the presence, in the low-frequency range studied here, of phase-shift effects observed (qualitatively) at higher frequencies [ 14,16 1.

Acknowledgements

We acknowledge E. Moran, I. Rasines and coworkers for providing us with the different high quality polycrystalline YBCO samples used in this work. This work has been supported by the Comision Interministerial de Ciencia Y Tecnologia (MAT 88-0769 and Mat 8%0250-CO2-01) and by the Programa para la Movilizacion de la Investigation, Desarrollo y Aplicaciones de la Superconductividad (Gran no. 89-3800), Spain. References [ 1 ] See, e.g., M. Tinkham, Introduction to Superconductivity, (McGraw-Hill, New York, 1975) chap. 2. [2] See, e.g., A.C. Rose-lnnes and E.H. Rhoderic, Introduction to superconductivity, Int. Series in Solid State Physics, vol. 6 (Pergamon, New York, 1978) chap. 1. [ 31 See, e.g., N.F. Mott and E.A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon, Oxford, 1979).

[4] See, e.g., M. Maglione, R. BBhmer, P. Lunkeuheimer, M. Lotze and A. Loidl, Physica Cl 53-155 (1988) 649. [ 51 See, e.g., C.Y. Chen, N.W. Preyer, P.J. Picone, M.A. Kastner, H.P. Jenssen, D.R. Gabbe, A. Cassanho and R.J. Birgeneau, Phys. Rev. Lett. 63 ( 1989) 2307. [6] See, e.g., J.C. Phillips, Phys Rev. B 38 (1988) 5019. [7] G.A. Samara, W.F. Hammetter and E.L. Venturini, Phys. Rev. B 41 (1990) 8974. [8] See, e.g., D.W. Deis, J.R. Gavaler, C.K. Jones and A. Patterson, J. Appl. Phys. 42 ( 197 1) 2 1. [ 9 ] See, e.g., A.M. Campbell and J.E. Evetts, Adv. in Phys. 2 1 ( 1972) 294, chap. 5, and references therein. lo] A. Veira, J. Maza, F. Miguelez, J. Ponte, C. Torron, F. Vidal, F. Garcia Alvarado, E. Moran, E. Garcia and M.A. Alario, J. Phys. D 21 (1988) 378. 111 J.A. Veira, G. Domarco, J. Maza, F. Migutlez, J. Ponte, C. Torron F. Vidal, J. Amador, M.T. Casais, C. Cascales, A. Castro, M. de Pedro and I. Rasines, J. Less Common Met. 150 (1989) 285. 121 Y. Schlesinger and S. Havlin, Rev. Solid State Sci, 1 ( 1987) 301. 13 ] Y. Schlesinger, S. Havlin and 1. Flener, Physica C 153- 155 (1988) 641. 141 A. Behrooz and A. Zettl. Solid State Commun. 70 (1989) 1059. [ 151 M. Goretzki, H.W. Helberg and K. Winzer, Synth. Metals 29 (1989) F569. [ 161 W. Ho, P.J. Hood, W.F. Hall, P. Kobrin, A.B. Harker and R.E. DeWames, Phys. Rev. B 38 (1988) 7029. [ 17 ] M.A. Alario, E. Moran, R. Saenz-Puche, F. Garcia Alvarado, V. Amador, B. Barahona, F. Femandez, M.T. Perz-Frias and J.L. Vicent, Mater. Res. Bull. 23 (1988) 313; P. Monod, F. D’Ivoire, J. Jegoudez, G. Collin and A. Revcolevschi, J. Phys. (Paris) 48 (1987) 1369; P. Millan, M.T. Casais and I. Rasines, private commun. [ 181 J. Kirtley, Y. Imry and P.K. Hansma, J. Low Temp. Phys. 17 (1974) 247. [ 191 J.A. Veira and F. Vidal, Physica C 159 (1989) 468. [20] J.A. Veira and F. Vidal, Phys. Rev. B 42 ( 1990) 8748. [21] S.J. Hagen, T.W. Jing, Z.Z. Wang, JR. Horvath and N.P. Ong, Phys. Rev. B 37 (1988) 7928. [ 22 ] G. Weigang and K. Winzer, Z. Phys. B 77 ( 1989) 11. [23] P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60 (1988) 132; See also: Normal State Transport Properties in High-T, superconductors: Evidence for Non-Fermi Liquid States, and attempts at Calculation, Preprint Princeton Condensed Matter Group. [24] T.A. Friedman, J.P. Rice, J. Gianpirtzakis and D.M. Ginzburg, Phys. Rev. B 39 (1989) 4258; M. Hikita and M. Suzuki, Phys. Rev. B 41 (1990) 834; T.A. Friedman, J.P. Rice, J. Gianpirtzakis and D.M. Ginzburg, Phys. Rev. B 42 (1990) 6217. [25] See, for instance: J.C. Phillips, Physics of High-T, Superconductors, Ed. (Academic Press New York, 1989). [26] See,e.g., D.H. Wu andS. Sridhar, Phys. Rev. Lett. 65 (1990) 2074. [27] See, e.g., M.A. Dubson, S.T. Herbert, J.J. Calabrese, D.C. Harris, B.R. Patton and J.C. Garland, Phys. Rev. Lett. 60 (1988) 1061.