Fluctuations in quasi-2D superconductors under magnetic field The case of YBa2Cu3O7−δ

PHYSICA ELSEVIER

Physica C 248 (1995) 138-146

Fluctuations in quasi-2D superconductors under magnetic field The case of YBa2Cu307_ C. Baraduc

a,b, A. Buzdin c,,, j_y. Henry a, j_p. Brison d, L. Puech d

a CEA/D~partement des Recherches Fondamentale sur la Matibre Condens~e, SPSMS/LCP, 38054 Grenoble Cedex 9, France b Ecole Normale Superieure, LPMC, 24 rue Lhomond, 75231 Paris Cedex 05, France e IGA, Ddpartement de Physique, Ecole Polytechnique F~d~rale de Lausanne, CH-1015 Lausanne, Switzerland Centre de Recherches sur les Tr~s Basses Temperatures, CNRS, B.P. 166, 38042 Grenoble Cedex 9, France

Received 15 February 1995

Abstract Both theoretical and experimental analysis of the fluctuational magnetization of quasi-2D superconductors is performed. The obtained theoretical expression is valid in the Gaussian regime of fluctuations for arbitrary anisotropy and describes very well the presented experimental data on YBa2Cu30 7_ 8 while 2D or 3D scalings fail.

1. Introduction Strong anisotropy and a large value of the Ginzburg-Levanjuk parameter lead to pronounced fluctuational effects in the high-Tc superconductors. The effects of these fluctuations have been observed on both the thermodynamical (specific heat [1], magnetization [2]) and kinetic (resistivity [3-5]) properties of these materials. Their study is motivated by the fact that these fluctuations are very sensitive to dimensional effects, an important issue in these systems. These dimensional effects are directly related to the degree of anisotropy of the system, which increases from YBazCu30 7_ 8 (YBaCuO) to bismuth-based HTSC compounds. The latter are practically 2D superconductors whereas in the former, a crossover between a 3D fluctuational regime near Tc and 2D regime at higher temperature has been reported, mainly through an analysis of paraconductivity measurements [3-5]. For YBaCuO, a quantitative description of the crossover from a 2D to a 3D regime is not trivial (for the case of paraconductivity, see for example Ref. [4]). The main difficulty is that the understanding of the normal-state behavior of the resistivity is rather poor, which makes it difficult to give a reliable estimate of the fluctuational effects on this quantity (due to the problem of background extraction). In this paper, we have chosen to focus on an analysis of the magnetization. The normal-state behavior is expected to be simple: the normal-state susceptibility (Xn) is a constant. In contrast an analysis of the specific heat is difficult as the electronic contribution to the specific heat is small (of the order of a few percent) and the

* Corresponding author. Also at Center for Condensed Matter Theory, P.O. Box 55, Moscow 109518, Russia. 0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0921-4534(95)00204-9

139

C. Baraduc et al. / Physica C 248 (1995) 138-146

phonon contribution is not so well known. There are, however, other difficulties with magnetization measurements which will be discussed below. In Section 2 of this paper we derive a general expression for the fluctuational magnetization in the Gaussian regime of moderately anisotropic layered superconductors which we expect to be applicable to YBaCuO. It is valid above the critical temperature T~(H), and is obtained in the framework of the Lawrence-Doniach model [6]. To our knowledge this is the first time such an expression has been derived for the whole Gaussian fluctuation region for quasi-2D superconductors (the anisotropic 3D case has been considered in Ref. [7] and for a review of fluctuations in 2D superconductors see Ref. [8] and literature cited therein). In the Appendix we present a formula for the fluctuational contribution to the specific heat of quasi-2D superconductors. In Sections 3 and 4, we present our measurements of M ( H ) on a YBaCuO single crystal, and compare them with the theoretical predictions.

2. General formulation

We start with the Lawrence-Doniach (LD) [6] functional which gives an adequate description of layered superconductors with weak coupling between the layers [9]. In this paper we restrict ourselves to the case of a magnetic field perpendicular to the layers (i.e. in the c direction, H = (0, 0, Hz)): its effect on T¢ is then more pronounced and so easier to detect experimentally. We may then choose the gauge: All = 1 / 2 ( H × r), A z = O, and write the functional

FLD[]=Efd2r

(

b aJ~bnl2+~l~bnJ4+

h_~H (

2ie ~ 2 ) VII---~-cAII]~bn +/J~b~+l-~b~J 2 .

(1)

~b~ is the order parameter of the nth superconducting layer, and a = a ( T - T¢o)/Tco = a r with ~-= ( T Tco)/T~o. Near To0 the LD functional is reduced to the usual Ginzburg-Landau (GL) functional with an effective mass along the c direction: M = h 2 / 4 t d 2 where d is the inter-layer distance. The superconducting coherence length along c is ~ 2c(0) = h 2 / 4 M a and the dimensionless anisotropy parameter of the LD model which determines the crossover from the 3D to the 2D regime is [9] r = 2~2(0)/d 2 = h 2 / 2 M a d 2 = 2 t / o t . Above the superconducting transition, in the Gaussian approximation, the fourth-order term in Eq. (1) can be neglected. The fluctuational contribution to the free energy AFfl may then be written as [10] AFfl

e

FLD(~b)

e- ,.--T= Je- ~ D ~ 0 =

~rkeT

~ E. +------~'

(2)

where E~ are the eigenvalues of the equation h2 [ 2ie ~2 ~mm ~Vl[ -- --~ -All] ~n+t(~On+l+~On-l-Z~On)=E~qJn" Performing a Fourier expansion on the coordinate n and taking into account Landau quantization in the perpendicular direction we may write for E~ hleJH

E~= - - ( n

+ 1 / 2 ) + a r ( 1 - cos k z d ) ,

(3)

mc

with u = {n, kz}. Then the fluctuation contribution to the free energy in the quasi-2D case is Vk BTh

AFn(h, T) =

4rrz~l~(0)d

£

~r

~

"rrk BT

•

~r dZn=0 ]~ In a [ ( 2 n + 1)h + r(1 - cos z) + z] '

(4)

h = H/H~2(O) is the dimensionless magnetic field, He2(0) = 2 m c a / e h being the linear extrapolation of He2 at T = 0 from the initial slope at Too.

C. Baraduc et al. /Physica C 248 (1995) 138-146

140

The fluctuation contribution to the free energy at H ~ 0 may be obtained from Eq. (4) with the help of the rN+ 1/2 well known summation formula E c~. = o f ( n ) ~ 1-1/2 fr zt x )x d x . Then the integral can be cut into a sum of 1 1 integrals between n - ~ and n + ~ and finally be written as Vk BTh "rt no 1_ ~rT 4~r2~12(0)d f_ ' ~ d Z y o f 2 ½ d x l n a [ ( n + l / 2 + x ) 2 h + r ( 1 - c o s

AFn(H=O,T)=

z)+r] " (5)

Here n o is a cutoff which is determined by the conditions of applicability of the Ginzburg-Landau theory (i.e. n o ~ H~2(O)/H). Combining Eqs. (4) and (5) we obtain in a similar way to Ref. [8] a convenient expression for the free-energy difference which converges at large n, so that the summation may be performed up to n = ~ without any problem of cutoff:

AFn(h, T) - AFn(H = 0, T) VkaTh -

~

1

~ h(2n + 1 + 2x) + r(1 - cos z) + r dxf de In ½ -~ h(2n + 1) + r(1 - cos z) + z

4'rrZ~l~(0)d =

After performing the integration over z and x, the field-dependent part of the free energy may be written as AFfl(h, T) - AFn(0, T)

-

n ( N In 2"tr~l~(0)dVkBT h ..~ ~°~

q~(N)=N+~-p

q~(N+ 1) q~(N+ 1) q~(N) + lnq~(N + 1/2)

pZ ~o(N+ 1) - q~(N)

) -~-AT+-i~-(~- ~ - 1 ,

(6)

2,

with N= n + r/2h + p, and p = r/2h. Note that the above formula is only valid for ~-> 0. The magnetic moment may then simply be calculated by numerically differentiating Eq. (6):

M = - OF~OH

for T > To0.

In the quasi-3D case we may perform the expansion over 1 / p in Eq. (6) and obtain

AFfl( h, T) -AFfl(0, T) = (2( T 2a.r~l~(0)¢zn~ ° -~ n + l + 2 - - h

VTh3/29/2

-

) 3/2

"/" 13/2

2(

--~ n+-~]

~/

-

1 r) n+-~+2-- ~

.

(7)

The 3D regime is established near the critical temperature (~" < r) while the magnetic field is not too high (h << r). Note that Eq. (7) is another representation of the field-dependent part of the fluctuation energy for ~-> 0 comparing to Ref. [11]. In the 3D case for ~'/2h << 1 the magnetization is

3TH1/ZV~

~ [2

_

1~1/2]

M = 47r~lZ(0)~zH32/2(0 ) Y0= t ' ~ ( n + l ) 3 / 2 - 3 2 n 3 / Z - ( n + 2 !

],

(8)

and has a H 1/2 dependence. The sum can be computed numerically and is roughly equal to 0.0608. In the 2D case for high field (r/2h << 1) we obtain, putting p = 0 in Eq. (6), that the magnetization is field independent in accordance with [8]: T ]~ (n+l 2,tr~lZ(0)dHc2(0 ) = n In

M= --

where the sum is equal to 0.346.

~

0

n

n+l +Inn+21 -

-

) 1 , --

(9)

C. Baraduc et aL / Physica C 248 0995) 138-146

141

For T < T~o (~"< 0), the magnetic moment should be written as:

M(-I~" I) -- AM+ 2M(~= 0) - g (

I~"I),

(10)

where AM = M ( - I~ I) + M( I ~ I) - 2M0- = 0) can be calculated from Eq. (4): Vk~T

1

O

~

q~(p)2

A M = - 27r~:12(0) d He2(0) ~-~h _ ° In p ( p + 1 0 1 ) ~ ( e - 1 0 1 ) ' here, P = n + p + 1, and 0 = "r/2h. So the expressions (6) and (10) give a complete description of the contribution of Gaussian fluctuations to the magnetization in the whole range of temperature T > T~(H) and for any field. It may then be possible to compare these formulas with experimental data for YBaCuO.

3. Experimental results: magnetization Our sample of YBa2Cu307_ a is a large porous single crystal (150 mg) in which the oxygen content has been optimized to obtain the maximal critical temperature. In this case, the distribution of To0 is less sensitive to possible oxygen inhomogeneity and indeed the transition width measured by low-field AC susceptibility is very small: BT~0= 0.14 K. However, the recrystallization technique used for synthesizing the sample leads to the formation of green-phase inclusions (Y2BaCuO5). Electron-microprobe investigations reveal that these inclusions (about 10 Ixm in size) are randomly distributed (including orientation) in the sample, so that their contribution to M can be considered as isotropic. X-ray diffraction indicates a very small mosaic spread of the crystal ( < 1%), an approximate oxygen concentration of 6.92, and an inclusion of around 18% (by mass) of the green phase. This impurity phase is taken into account in our data analysis. Magnetization measurements were performed using a commercial SQUID magnetometer (SHE). The magnetic field (up to 4 T) was applied along the c-axis with an accuracy of 1 degree. The magnetic signal from the sample holder was measured and substracted from the data. We have also made a small correction ( < 1%) to the value of the applied field in order to obtain a perfect linear dependence of the magnetization with the field for temperatures between 100 and 102 K. This correction is within the error bars of the values specified for our magnetometer. Fig. 1 displays the raw data of the magnetization (after subtraction of the sample-holder contribution) for different applied fields, in the reversible region. A zero-field critical temperature (T~0 = 92.8 K) has been

10

85 , ....

-5 "?,

,~

90 ,.

~

o

o:i:1cll:lcll:Jc:moffli:ii:~clo

S ~l

o

o o v

o ~ ~

-15

T (K) • ,

a

o

-lO

95 100 v,,~w~vv~w.

1r

2T 3T 4T

o

-20 -2~

~

-0.1

=

= '~'

J

. . . .

-0.05

I

. . . .

I

. . . .

0 0.05 IT/To0 -11

=

0.1

. . . .

0.15

Fig. 1. Raw data of the magnetization M of our sample (in emu), for fields of 1, 2, 3 and 4 T as a function of the reduced temperature. T~0 = 92.7 K is the zero-field transition temperature.

C. Baraduc et al./Physica C 248 (1995) 138-146

142

obtained from low-field AC susceptibility measurements. The measured magnetization is composed of contributions from: (1) the normal-state susceptibility of YBaCuO (X.). This is assumed to be constant because A X = X c - X,b does not depend on temperature. A X corresponds indeed to the differential susceptibility of pure YBaCuO since the distribution of green phase is isotropic. Moreover yttrium Knight shift and T1 relaxation-time measurements [12] show that the sample is at the optimum doping. And in that case the susceptibility of YBaCuO is known to be constant. (2) The diamagnetic magnetization M n due to the Gaussian fluctuations. (3) The Curie-Weiss susceptibility of the green-phase Xcw which clearly shows up in the data through the upward curvature of M. The values of Xn and Xcw are known from the literature: Xn = 6 × 10 -7 e m u / g [13,14], Xcw = C/(T+ O) with C = 9.24 × 10 -4 emu K / g and O = 49 K [15], or C = 8.74 × 1 0 - 4 e m u K / g and 19 = 39 K [16]. Calling x the mass ratio of the green phase, and mtot the (measured) total mass of our sample (into t = 0.150 g), we can write M

=

mtot{(1 -- x ) ( Xn H

+ Mr) + xHC/(T + 19)}.

(11)

Taking as a starting point x = 18%, we can estimate that for T ~ 100 K the green-phase contribution is approximately 50% of the total; the other 50% is mostly Xn" AS x is not known precisely and the green-phase contribution to M is large, we had to leave this contribution as a free adjustable parameter. The values of C and 19 were fixed (averaged values of the literature), as were the values for Xn and mtot. M r w a g deduced from a numerical estimate of Eqs. (6) and (10). In these formulas, the prefactor VkBT 2'n'~l~ (0) d was rewritten as

- x ) kaT

mtot(1

PYBCO

~ o dHc2(O)'

.4..,.,l,.,.I.,.,,,.,.I,1.,.,.

|.l.l,l,l.|,l.l.l.l.l,l,l.,l|.l,l,l,l,l,|,l,l,l,|,l.I

t- 2.2

~'

2

N ~.8

vo

DA ,

a

1.6

I,

I

,

I

0

•

I

l

I

,

I

.

I

•

0.04

I

•

I

,

I

,

I

.

I

0.08

.

I

I , l . l , l l l .

0

I.

4T fit

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

0.04 0.08 ( T / T 0 -1)

o

oA

13 I , l , l , l ,

0

I,

I . I l l , | . l , l . l . I

0.04

0.08

Fig. 2. Best fits of the data plotted as M / H vs. reduced temperature for different values of the anisotropy parameter r = 2~e2(0)/d 2 of the Lawrence-Doniach model. For each value of r, x (the green-phase mass ratio) and 0 H e 2 / a T (initial slope at Tc) are adjusted. Values of x = 18.95%, 18.5%, and 18.2% as well as aHc2 lOT = 1.314 T, 1.314 T and 1.36 T / K are found for r = 0.04, 0.057 and 0.07, respectively.

C. Baraducet al./ Physica C 248 (1995)138-146 90

92

94

96

98

143

100

T (K

102

-10

-15

•0 O

~

¢

=

;~

o

2T 1

~_~it r

-20

~ /

-25

r=0.057 0Hc2/!T=1.314 18.5% of green phase

v

"30

I

.

-0.02

I

'

I

0

,

I

,

I

0.02

,

i

i

I

0.04

,

I

t

i

0.06

,

i

.

I

0.08

•

i

.

I

0.1

,

I

.

0.12

(T/To0 - 1) Fig. 3. Fluctuational contribution to the magnetization of YBaCuO (after subtraction from the data of the green-phase and normal-state contributions), together with the best fit obtained for r = 2~2(0)/d 2 = 0.057 and 0//c2lOT = 1.314 T/K.

where RYBCO 6.4 g / c m 3 is theo mass density of pure YBaCuO, ~0 is the flux quantum, and the inter-layer distance d was fixed at d = 12 A, the unit-cell parameter. We are left with only three parameters to fit the four curves of Fig. 1. (1) the mass ratio of the green phase, (2) Hc2(0), the upper critical field along the c-axis extrapolated to T = 0, and (3) r = 2 t / t r , the anisotropy parameter of the Lawrence-Doniach model. Using an algorithm based on the Levenberg and Marquardt method [17], we have calculated, for fixed values of r, the best values of x and HOE(0) for the 4 fields of Fig. 1 simultaneously. Keeping in mind that formulae (6) and (10) are only valid in the Gaussian regime, for each field H, we only fitted data for ~"> (H/HoE(O) + 0.01): the critical region is assumed to be less than 1% of T~0 for all magnetic fields in our experiments. Fig. 2 displays the results of these fits (plotted as M / H versus ~') for different values of r. The best fit was found for r = 0.057, HOE(0) = 121.8 T (i.e. an initial slope at T¢0 of ( - d H ~ E / d T ) = - 1.31 T / K ) andx = 18.5%. The fit of Mfl alone (after subtraction of the contributions of Xn and X~w deduced from our calculations) is shown in Fig. 3. =

4. Discussion

Usually the analysis of the fluctuational magnetization is done by looking for scaling laws. The advantage of these methods is that they are valid in both the Gaussian and critical regimes. The limitation comes from the fact that models exist only for the asymptotical regimes (purely 2D or 3D), so that a quantitative study of the crossover is difficult. Moreover, in the case of YBaCuO, even if the experimental data satisfy some scaling [18], the theoretical expression for the scaling function does not fit the data [19] because the dimensionality is supposed to be fixed, either purely 2D or 3D. So these theoretical expressions cannot account for fluctuations in moderately anisotropic systems such as YBaCuO. Therefore it is very difficult to gain quantitative information by such analysis.

144

C. Baraduc et al. /Physica C 248 (1995) 138-146 94

96

98

100

102

96 104 112 120 128 136 144 T(K)

0

.°°

,/

,,"

~.-6 [- ~

{

,"

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

I,"t; "

I-lll

[ ....... 3o

[.

-10

0

o.o2 o . o 4 o . o 6

0.08

(T/Too -1)

0.1

0

0.1

o.2

o.3

o.4

o.s

0.6

o.7

(T/T©o -1)

Fig. 4. Left-hand side: comparison of the experimental fluctuational contribution to M for H = 1 T and H = 4 T with the pure 2D and 3D regimes. Same values of 0He2 ~aT = 1.314 T / K as well as d = 12 ~,, r = 0 for the 2D case, and ~c(0) = d ~ / ~ = 2.03 A (from r = 0.057, d = 12/~) for the 3D case. Right-hand side: comparison between the pure 2D, 3D formulas and the theoretical expression for r = 0.057 in a wider temperature range.

Although it is restricted to the Gaussian regime, the main advantage of our model is that a fit of M n is made with absolute units, and accounts well for the data. The value that we have found for r ( r - - 0 . 0 5 7 ) clearly shows that, as expected, YBaCuO is in an intermediate regime between 2D and 3D. This value corresponds to a coherence length along the c-axis of ~c(0) = 2.03 ,~. Note that the value r = 0.05 + 0.02 has been obtained previously from the analysis of parallel and perpendicular paraconductivity [4] within the same LawrenceDoniach model. Fig. 4 shows, for the minimum and maximum fields of our experiment, a comparison between purely 2D or 3D formula and the data: it can be seen that none of these functions can reproduce satisfactorily the observed behavior. This explains the failure of the "scaling-law approach". We conclude, however, that in the temperature and field range of our experiment, YBaCuO is much closer to a 3D superconductor than a 2D one. The value of OHc2/aT=-1.31 T / K is in agreement with what is reported from resistivity [20] or specific-heat [21] measurements, which yield values between - 1 . 4 and - 1 . 8 T / K . It is always difficult to define and measure precisely He2 in this compound because of the width of the transition under a field. The best would have been to compare directly the value of these parameters with an analogous fit of the specific-heat results, but the broadening of the transition in a magnetic field as observed on the specific heat prevents any quantitative comparison with our model. Nevertheless, we also derive in the Appendix a formula valid for the fiuctuational contribution to the specific heat in the Gaussian regime. In conclusion we have found that the proposed theoretical expression for the fluctuational magnetization of quasi-2D superconductors accounts remarkably well for our experimental data on YBaCuO. It is possible to fit simultaneously all the experimental curves with only three well controlled adjustable parameters: the green-phase mass ratio, the slope of the upper critical field and the anisotropy. Both are in good agreement with other experiments (like resistivity). Finally, let us emphasize that YBaCuO is truly an intermediate case as regards the dimensionality of the superconductivity: neither a pure 2D nor a pure 3D scaling can describe the fluctuational magnetization.

145

C. Baraduc et al. / Physica C 248 (1995) 138-146 Acknowledgements

We thank A. Carrington for critical reading of the manuscript and useful remarks. We would also like to thank the INTAS (Ref. No. 2452) and "Fonds National Suisse de la Recherche Scientifique" (FRPNR30 No. 4030-32794) for partial support of this work.

Appendix

Fluctuational contribution

to

the specific heat

Using the representation (6) for the free energy we may derive a useful formula for the fluctuational contribution to the specific heat 02F

3S

=T--. OT

C = - T-~

At temperatures above T¢ the difference between specific heat with and without magnetic field is Cfl(H, T ) - Cfl(0, T)

r

2-trY

x-" (

Tc4"tr2d~l~ ~'n l+r[

In

q~(N+ 1)

1

((N+1)2_p2 1

1

+ 4--

2

2 + ((N+l)Z_pz)

]}

3/2;

q~( N ) = N + f N S - p 2 ,

with N = n + 7 / 2 h + p, and p = r / 2 h . Finally to compare with experiment it may be useful to have the expression for difference between fluctuation contributions at different fields: Cfl( H ,, T ) - Cfl(H2, T)

V

T l+'r +

4h,

NI+ ~ ((N1 + 1 ) 2

_

I+T 2~3/2 Pl)

4h 2

_N2+ - 1\ 2 2'I3/2 I/ ' ((N2 'l- T) - P 2 J ]/

where N1, 2 = n + r / 2 h i , z + PL2, and Pa,2 = r/2ha,2 and the logarithm term is due to the different cut-off nl, o = H c 2 ( O ) / H 1 at H = H 1 and n2, o = HcE(O)/H z at H = H 2 and the r e m a i n i n g s u m m a t i o n m a y be e x t e n d e d to n - - o o .

References

[1] A. Junod, in: Physical Properties of I-ITSC, vol. II, Thermodynamics of HTSC, ed. D. Ginsberg (World Scientific, Singapore, 1990). [2] W.C. Lee, R.A. Klemm and D.C. Johnston, Phys. Rev. Lett 63 (1989) 1012. [3] A. Kapitulnik, Physica C 153-155 (1988) 520.

146 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

C. Baraduc et al. /Physica C 248 (1995) 138-146

C. Baraduc, V. Pagnon, A. Buzdin, J.Y. Henry and C. Ayache, Phys. Lett. A 166 (1992) 267. J.A. Veira and F. Vidai, Phys. Rev. B 42 (1990) 13. W.E. Lawrence and S. Doniach, in: Proc. 12th Int. Conf. on Low-Temp. Phys. ed., E. Kanda (Kyoto, 1970) p. 361. A. Buzdin and D. Feinberg, Physica C 220 (1994) 74. A.E. Koshelev, Phys. Rev. B 50 (1994) 506. L.N. Bulaevskii, Int. J. Mod. Phys. 134 (1990) 1849. E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics, vol. 9 (Pergamon, Oxford, 1980). R.E. Prange, Phys. Rev. B 1 (1970) 2349. M. Horvatic and C. Berthier, private communication. W.C. Lee and D.C. Johnston, Phys. Rev. B 41 (1990) 1904. Y. Yamaguchi, M. Tokumoto, S. Waki, J. Nakagawa and Y. Kimura, J. Phys. Soc. Jpn. 58 (1989) 2256. S.S.P. Parkin, E.M. Engler, V.Y. Lee and R.B. Beyers, Phys. Rev. B 37 (1988) 131. G. Triscone, PhD. thesis (Gen~ve 1993). W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vettering, Numerical Recipies in C (Cambridge Univ. Press, Cambridge, 1988). U. Welp, S. Fleshier, K.W. Kwok, R.A. Klemm, V.M. Vinokur, J. Downey, B. Veal and G.W. Crabtree, Phys. Rev. Lett. 67 (1991) 3180. [19] N.K. Wilkin and M.A. Moore, Phys. Rev. B 48 (1993) 3464. [20] D. Racah, A. Kapitulnik and G. Deutscher, Physica B 194-196 (1994) 1799. [21] A. Junod, E. Bonjour, R. Calemczuk, J.Y. Henry, J. Muller, G. Triscone and J.C. Vallier, Physica C 211 (1993) 304.