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PHYSICAL REVIEW B VOLUME 47, NUMBER 13 1 APRIL 1993-I
Lower critical fields, critical currents, and fiux creepin PbzSr2R t „Ca„Cu3Os+y (R =Y,Dy) single crystals
V. V. Metlushko* and G. Guntherodt2. Physikalishes Institut, Rheinisch We-stfiilische Technische Hochschule Aachen, Templergraben 55, W 510-0Aachen, Germany
V. V. Moshchalkov' and Y. BruynseraedeLaboratorium uoor Vaste Stof-Fysika en Magnetisme, Katholieke Uniuersiteit te Leuven, Celestij nenlaan 200D,
B-3001 Leuven, Belgium
M. M. LukinaPhysics Department, Moscow State University, 117234, Moscow, Russia
(Received 8 October 1992)
The lower critical field H, &( T), temperature and field dependences of the critical current j, ( T,H), andthe normalized magnetization relaxation rate d (lnM)/d (1nt) have been studied for two orientations ofthe magnetic field H~~c and Hlc in Pb2SrzR, „Ca„Cu308+~ (R =Y,Dy) single crystals having practicallya cubic shape with characteristic dimensions 0.6X0.6X0.5 mm . H„(T) shows for H~~c a positive cur-vature with decreasing temperature without the usual low-temperature saturation found for the H|corientation. This anomalous H~~ l( T) behavior is related to the modification of the character of the fieldpenetration in a layered structure consisting of a stack of superconducting and nonsuperconductingplanes. The temperature dependence of the normalized relaxation rate S =d (1nM)/d (lnt) shows a max-imum at temperatures where the largest anisotropy of the critical currents is observed. Magnetic mea-surements have been used to obtain current-voltage characteristics from the damping electric field deter-mined by the magnetic-flux creep rate. The analysis of j,(T,H), the temperature and field dependencesof d(lnM)/d (lnt), and of the H, &(T) data shows that a continuous decoupling of the pancakelike vor-tices takes place with increasing temperature or magnetic field. A striking similarity between the tem-perature dependences of the normalized relaxation rate observed for H~~c and the H orientation veryclose to Hlc implies that the flux-phase dynamics is mainly determined by the component of the appliedmagnetic field normal to the (a, b) planes, i.e., by the thermal activation of the motion of the pancakevortices in Pb&Sr2R i „Ca Cu308+y This material is of particular interest, because its anisotropic be-havior is intermediate between that of YBa2Cu307 and Bi2Sr2CaCu2OX and its T, = 80 K is comparableto that of Bi2Sr2CaCu20 .
I. INTRODUCTION
Among other high-T, oxides, the recently discoveredfamily of superconductors PbzSr2R, „Ca,Cu30, +s (Ref.1) has not been studied in detail yet, which is mainly re-lated to difficulties in obtaining relatively large high qual-ity single crystals. As a result, only a few reported inves-tigations have been performed so far on single crys-tals. Reedyk et ah. " studied tem. perature dependences oflower critical fields H, &( T) and London penetrationdepths L (T). The characteristic scale for the flux-creepactivation energy Uo =20—90 meV was obtained byPradhan et al. on a Pb2Sr2 Y
& Ca„Cu308+ & singlecrystal with T, =35 K, which is noticeably smaller thanthe highest possible T, for this system, typically about 80
Using the Aux growth method from the non-stoichiometric melts with an excess of PbO, SrO, andCuO (Ref. 2), we were recently able to obtain quite largehigh quality single crystals having an almost perfect cu-bic shape. To grow the crystals, the crucible was quicklyheated up to 1030 C, held at this temperature for 30 minand then cooled to 800 C at 5 C/h and to 550'C at
20'C/h. Then the crucible was cooled in the furnace in aHow of N2 gas. The single crystals were carefully extract-ed from the Aux and the composition determined by thex-ray microprobe using a JSM-820 Jeol electron micro-scope with an accuracy of +0. 1%%uo. All measurements re-ported in the present paper have been carried out on thePb2 ppS12 ppRp 67 Cap 33CU3 ppOs+s (R =Y,Dy) single crys-tals with T, =80 K and characteristic dimensions of0.6X0.6X0.5 mm . We have studied temperature, time,and field dependences of the magnetization M on aMPMS2 Quantum Design SQUID magnetometer withthe scan length 3 cm, corresponding to a field homogenei-ty of better than 0.05%.
II. LOWER CRITICAL FIELDS
As we have already mentioned, an almost cubic shapeof the available single crystals makes them very suitablefor studies of the anisotropy of different superconductingproperties. The demagnetizing factor nd calculated inthe approximation of an inscribed ellipsoid ' for oursamples is nearly the same for both H~~c(nd =0.371) andHlc( nd =0.315 ) orientations. This diff'erence in the
47 8212 1993 The American Physical Society
47 LOWER CRITICAL FIELDS, CRITICAL CURRENTS, AND. . . 8213
demagnetizing factors nd (H~~c) and nd (Hlc) is muchsmaller than that for typical plateletlike high-T, crystals.
Temperature dependences of the magnetic momentP =MV of Pb2Sr2Y, Ca Cu3O8+&, where M is themagnetization and V is the volume of the crystal, aregiven in Fig. 1 for H = 10 G. We note that all magneticdata are presented throughout this paper in P vs T, Hplots rather than in M vs T, H curves to exclude an addi-tional error in M appearing because of uncertainties inthe volume determination. The P vs T curve (Fig. 1)shows a sharp superconducting transition with the sametransition temperature T, =80 K for the two systemsPb2Sr2Y, „Ca„Cu30s+s (PSYCCO), see Fig. 1 andPb2Sr2Dy, Ca Cu30s+s (PSDCCO), see Fig. 2(a). Thelower the applied magnetic field, the sharper the super-conducting transition. As in all other known high-T, ox-ides, the saturation value of P at T~O is much smallerfor the Meissner field-cooled (FC) than for the shieldingzero-field-cooled (ZFC) measurements.
The magnetization of single crystals with Dy consistsof two contributions arising from the diamagneticresponse of the superconducting state and the paramag-netic response form the rare-earth Dy + ions. The latteris strongly field and temperature dependent. In highfields and at lower temperatures, the paramagneticresponse dominates over the diamagnetic one as shown inFig. 2(b), where the resulting P" (T) curve measured forH =535.6 G lies (for T ~ T, ) completely in the paramag-netic region [Fig. 2(b)]. The paramagnetic componentcan be fitted by a Curie-Weiss dependenceM =c onst/( T+0) with a parameter 0= —3 K in quali-tative agreement with the effective magnetic moment ofan isolated Dy + ion.
A detailed analysis of the P" (T) data is presented inFig. 3 where a strong field dependence of the Meissnerfraction is demonstrated for both H~~c and Hlc fieldorientations. The effective magnetic field H,z has beencalculated by taking into account the demagnetizingfields.
The drastic suppression of the Meissner fraction by anapplied field implies that in order to obtain the full 100%Meissner e8ect, very low fields (less than 10 G" '
)
2
IC) -4-T
ECL -6—
I
~ooooa&o o oooo ~FC G
800
E
O 4o
ECL
+
++
~+Cl +~+
(b)
8
530
I
C)GD
3 3C
I2~ZFC ~ CP
H 5.4G, H~ cg
I I
60 80 100I I
0 20 40 0 20 40 60 80 100T(K}
FIG. 2. (a) Temperature dependence of the magnetic momentP of a Pb2Sr2Dy& „Ca Cu30$+y single crystal(H~~c): =ZFC, Q=FC. (b) Temperature dependence of Pand (dP "
)' of a Pb2Sr2Dy &
Ca Cu308+y single crystal,H=535. 6 Cx, H~~c.
where H*=mj, d/5 for a slab and H'=4~j, A/10 for acylinder and 0~ m, ~ 1. The extrapolation method maybe used in this form when the field dependence of critical
must be used.The magnetic moment vs field curves measured at
different fixed temperatures have been used to determinethe lower critical field. For this purpose we subtractedfirst the ideal linear contribution corresponding to thefull diamagnetic response. Then the resulting difference6P is analyzed by the extrapolation method' ' which isbased on the simple behavior 5P —(H —H„), useful forpractical purposes in finding H„as the field where thelinear (5M)'~ vs H dependence intersects the H axis ofthe (5P )' vs H plot (Fig. 4). For a slab with thicknessd or a cylinder with radius A, the following relation be-tween 5M and H is valid'
(H —H ) (H —H )Mf= +2(H* —H„) 4(H* —H, i)
2H, i(1 —m, q)(H —H, i)+(H* —H, i)
E-
IEO
D
g$ ~
0—
10-15-20-25-30-
H106ZFC, H ll cZFC, HJ cFC, HllcFC, Hic
0.4
0.3
' 0.2
0.1
Hll cHgc
T 5K
-350
I
20I I
40 60Temperature (K)
80 ~000-0
I
20I
40H "'(G)
60 80
FIG. 1. Temperature dependence of the magnetic momentP of a Pb2Sr2 Y, „Ca Cu308+y single crystal, H = 1O G:
=ZFC, Hiic, 6 =FC, Hiic, Q' =ZFC, Hlc, O =FC, Hlc.
FIG. 3. Field dependence of the Meissner phase of aPb, Sr,Y, „Ca„Cu30&+~ single crystal at 5 K: =H~~c,6 =Hlc.
8214 V. V. METLUSHKO et aL
(6p (emu))0.08
0.06—
0.04—
.5K5K
5K
H (G)
600
500- g
400—
0.8
0.4
0.2
H, /H (0)
0.02—300— 0
o o.2 0.4 o.e o.eT/Tc
llc
0.000 200 t400 600 800 1000 1200 1400 1600
H„ H ~«(G)
200-
HJc
00 ooo o
Hl[c
1/2( SP (emu))
0.10T~20K
oo
0 9 g ~ n ~, o
0 10 20 30 40 50 60 70
0.08—
0.06—
0.04—
0.02—
~ ~
~ ~
~ ~
TI35KT*40K
~ ~R
~ ~
~ ~~ ~~ ~
~ ~4
30K T 25K
Hll c
FIG. 5. Temperature dependence of the low critical, field H, &
of a PbzSrzY, „Ca„Cu&O,+~ single crystal: =H~~c, 0 =Hlc(A =H ~~c and 0=Hlc data taken from Reedyk et al. in Ref. 4).Inset: H, &( T) curves calculated for S-N-S multilayers withdift'erent weight coeKcient o of N layers (curve 1 —o = 5,2 —o.= 10, 3 —o.= 100) (data taken from Cxolubov et al. in Ref.28).
0.000
I~ ~ ~
200 400 600 800 1000 1200 1400H "'(G)
Flax. 4. 6P'~~ vs the demagnetization-corrected field H, sH~~c
for Pb2Sr, Y& Ca Cu308+y
gives the effective-mass ratio m, /m, b 960, indicating awell-defined regime of a weak coupling between super-conducting Cu02 bilayers along the c axis. The quasi-two-dimensional (2D) character of high T, oxide-s has
current j,(H) is negligible and the second term in Eq. (1)is much smaller than the first one. As it is illustrated inFig. 4, a good linearity of (5M)'~ vs H,s is clearly seenfor H~~c in a wide range of fields and temperatures. ForHlc, the (5M)'~ vs H, s curve has a more complicatedshape, indicating the importance of the second term inEq. (1).
Figure 5 shows temperature dependences of lower criti-cal fields obtained by the extrapolation method. Weshould note here that our H, ;(T) data give the highestpossibie estimate of H, '„which means that the real H, ',
values might be even lower. For the H~~c orientationthere is a good agreement of our H,";(T) data with thosereported earlier by Reedyk et al. which are alsodisplayed in Fig. 5, with H,"'i(0)=500 G. For the Hlccase, we have obtained systematically lower H„values,compared to Ref. 4, with H, ;(0)=16 G. The anisotropyparameter
I' —(m /rri )1/2 Hllc /Hlc —I /L, ) 31
been emphasized before for Bi2Sr2CaCuzO~(I =50—140), ' ' T12Ba2CaCu20 (I =70), ' and forYBa2Cu3O, ( I"= 5 ). '
The anomalous temperature dependence of H,";(T) isalso a signature of the layered structure of high-T, ox-ides, which may be effectively treated as S-N-S or S-I-Smultilayers with the superconducting Cu02 layer separat-ed by nonsuperconducting layers. The contributionarising from these layers leads to an appearance of a posi-tive H, i( T) curvature at low temperatures. We havemade a qualitative comparison of our measured 0,"', (T)curve with the model calculation for the S-N-S multi-layers (see inset of Fig. 5). The theoretical fit may de-scribe an anomalous positive curvature of H,"', ( T), thoughthe constraints used in Ref. 28 might not be suitable forthe case of high-T, materials, where the space layers maybe insulating.
Due to ambiguities in the determination of H, ;( T), weare not discussing the H, i( T) dependence given in Fig. 5
and also its difference with the data reported by Reedyket al. Nevertheless, we are sure that our data give onlyan upper limit for H, ;( T), i.e., the real anisotropy may be,in fact, even essentially higher than I =31 mentionedabove.
Having an anisotropic (I ~ 31) superconducting sam-ple with practically a cubic shape, it was interesting tocheck experimentally the theoretical calculation forthe field dependences of the magnetic moment fordifferent angles between H and the c axis. Plotting the re-versible part of the magnetic moment for Hlc vs H (Fig.6), we clearly see the presence of the two maxima, in
LO%'ER CRITICAL FIELDS, CRITICAL CURRENTS, AND. . . 8215
REV -4-P (10 emu)
REV
Pm (10 emu)0
2.5— T=30KHzc
-10-
H/H ,c10.80 -30
70
30-40-
o o
0.00
10
10H/H cl
I
20
85
90
I
3Q 4QH(kG}
o-50 I I I I I I III I I I I I I I II I I l I I I II I I I I I I I II
10 100 1000 10000 100000H(kG)
FIG. 7. Field dependence of the reversible magnetic momentP" of a Pb2Sr2Y~ „Ca,Cu30, +~ single crystal, H~~c, T=65K.
FIG. 6. Field dependence of the reversible magnetic momentP" of a PbzSrzY& Ca Cu3O8+ single crystal, H near Hlcdirection, T=30 K. Inset: Full magnetic moment M vs exter-nal magnetic field H for various angles 0, in an ellipsoidal sam-
ple of a layered superconductor (data taken from Buzdin et al.in Ref. 30), where 0, is the angle between the c axis and thefield direction.
Joseph son-coupled quasi-2D layered superconductorspredicts for the H orientations not too close to H~~ab asimilar relation for the magnetization component perpen-dicular to the (a, b) plane:
IIqH, ~ln
Hll~ab
+CX2D (5)
correspondence with the theoretical P (H) data for0„=85', i.e., for small tilt angles u=(m'/2) —0, =5,where 0, is an angle between the vortex lattice and the caxis. From this point of view, the real H orientation inour experiment seems to deviate by an angle of a-5'from the Cu02 planes. For tilted fields this results in '
Hll',H, &
"measured" =H, ;+cosO.
and the difference between our estimates for H, ; (Fig. 5)and Reedyk's data may be easily explained as originatingmainly from a small misorientation, which, however,might be very important for strongly anisotropic super-conductors.
Field dependences of the magnetic moment P (H) canbe decomposed into two parts corresponding to reversible(P ) and irreversible terms (P' ):
where e2D is a vortex core contribution and g is a con-stant of order l for a hexagonal lattice. Using Eq. (4), wecalculated the temperature variation of L,b, which can bededuced from Fig. 7. By plotting P vs lnH, we see thatthere is a wide range of fields where Eqs. (4) and (5), pre-dicting a P vs lnH relation, are fulfilled. The slopedP /d(lnH) gives L,b (T) (see Fig. 8), which follows aBCS-like behavior (solid line in Fig. 8 and dashed lines inFig. 9). The difference between our data (open circles,Fig. 9) and those reported by Reedyk et al. (opensquares, Fig. 9) is eliminated after the proper normaliza-tion by the different T, values (T, =80 K, Fig. l andT, = 75 K, Ref. 4). We have obtained the zero-
(L ab (p, m))6
P (H)= —,'[P+(H)+P (H)],P' "(H)=
—,' [P+(H) —P (H)], (3)
Hll'ln
32~2L 2H Ill
(4)
where P (H) and P (H) are the upper and lowerbranches of hysteresis loops, respectively. The reversiblecontribution arises from the vortex lattice magnetizationand it can be used to derive the temperature dependenceof the penetration depth L. The Ginzburg-Landau (GL)analysis gives for the anisotropic 3D superconductors thefollowing relation:
00.75 0.85 0.95 TZT,
where N is the Aux quantum, L,b is the a, b-planepenetration depth, and P is a constant. The theory of the
FICx. 8. L,b vs normalized temperature forPb, Sr,Y, „Ca„Cu30&+~, H~~c where L,I, is the penetrationdepth of the a, b plane.
47V. V. METLUSHKO et al.8216
j (A/cm )
)06
HiIc Ref.4
Hl. c Ref.4
H II c
I
I
IIt
Q
0
i
y
100
Hll c, PSYCGQ
H&c, PSYCGO
Hll c, PSDGGO
10200
20I
600 I I I I I l 1
63 65 67 69 71 73 75 77 79T(K)
L( m)
Hll c Ref.44 ~ Hzc Ref 4
Hltc
II
II
I
A/
/9
V
00.8
I
T/Tc
0.9 0.950.85
FIG. 9. Temperature dependence of the penetration depth Lof a Pb2SrzY, „Ca„Cu,O, +» single crystal o =H~~c (H=H~~cand D =Hie data taken from Reedyk et al. in Ref. 4).
temperature penetration depth in the ah planeL,b(0) = 1860 A, which is in agreement withL,b(0)=2575 A (Ref. 4) and not far from the L,b(0)value in YBa2Cu30 [L,~(0)=1300—1700 A (Refs. 14,38, and 39)] and Bi2Sr2CaCu20» [L,&(0)=3000 A (Ref.40)].
III. CRITICAL CURRENTS
The irreversible part of the magnetization [Eq. (3)]determines the behavior of the critical currents j,. ForH~~c, the correspondence between j, and the width ofhysteresis loops is given by the isotropic Bean model,
0 40 80
FIG. 10. Temperature dependence of the criticalcurrent j, at H=O for: =Pb2Sr2Y& „Ca„Cu308+y,H~~c, A=PbzSrzY, Ca, Cu30s~», HLc,+ =PbzSr2Dy, ,Ca„Cu,O, +», H
~~c.
whereas for Hlc the derivation of j, should be based onthe anisotropic Bean model. This procedure has beenused to find the j,(H, T) dependences. Figure 10 showsthe zero-field temperature variation of j, of PSYCCO forboth H~~c and Hlc orientations and compares it withPSDCCO for H~~c. The j,(T) curve is characterized by aquasi-exponential dependence observed previously in oth-er high-T, oxides (see Fig. 11). However, there is a no-ticeable change of the j,(T) slope around T"=15—20 K(T/T, =0.25) which occurs continuously in YBa2Cu30[Fig. 11(a)] and in Pb2Sr&R1 Ca Cu30s+» (Fig.10), but abruptly in Bi2Sr2CBCu20 [Fig. 11(a)]. Aswas suggested before, this type of anomaly seems to be
S -Mo (dM/dint)0.12'
j,&],(0)
~- BSCCO
Y BOO
BK BO~ 1SVCCO0.10-
0.08I
BKBO
100F
PSYCCO
(b)
-1-10 0.06-
0.04[ p .
s~
P.
-2-10
0.02
-sr10 02 04 06 T/T 1.0
C0 0.2 0.4 0.6 T~ T 1.0
C
FIG. 11. (a) Temperature dependence of the critical current:=Bi,Sr,CaCu20„(Ref. 44), 0=YBa2Cu, O» (Ref. 41),
A=Ba, I| Bi03 (Ref. 54), o =Pb,Sr,Y& XCa Cu30, +y. (b)The temperature dependence of the normalized relaxation rateS: =BizSrzCaCu20„(Ref. 44), 0=YBa2Cu30» (Ref. 41),0 =Ba, K BiO, (Ref. 54), o =Pb,Sr,Y, ,Ca„Cu308+y.
47 LOWER CRITICAL FIELDS, CRITICAL CURRENTS, AND. . . 8217
j (A/cm21
10
105 =
104
10
102
Pb2sr2Y1 „Ca„CU30
T 5K
OK
OK
induced decoupling of the pancake vortices may explainthe appearance of the so-called "peak effect" (Ref. 51) inhigh-T, oxides, when j, becomes larger with increasingfield, forming a fishtail shape ' ' or butterAylike hys-teresis loop.
The anisotropy of critical currents K =j,"/j, deter-mined from magnetization data for H =0 is slowly chang-ing with temperature [Fig. 13~a)] and it has a maximumK =12 at a temperature close to T". At the same tem-perature the normalized relaxation rateS =d(lnM)/d(lnt) [Fig. 13(b)] also exhibits a maximum.The details of the S(T) behavior will be discussed below.
IV. FLUX CREEP
10 01
20
11 C
H(kG)
FIG. 12. Field dependence of the critical current ofPbzSr2Y, „Ca„Cu30,+~, H~~c at various temperatures.
20
15-
10-
(a)
S M dM/d(lnt)
0.06—
(b)0.05—
0 0at-j~
0.04—
0
0.03- 0 0 0cj
0.02
0.01—
0 1 I I I
0 10 20 30 40 50 60 ?0 80T(K)
0 I
0 10 20 30 40 50 60 ?0 80T(K)
FIG. 13. (a) Temperature dependence of the critical currentanisotropy j, /j, for a Pb2Sr2Y& Ca Cu308+y single crystal.(b) Temperature dependence of the normalized relaxation rate5 =Mo 'dM/d(lnt) of a Pb&Sr&Y& Ca Cu3O8+y single crystal:O=H~~c, 4 =Hlc.
caused by the crossover from the classical Abrikosov vor-tex lattice with pinning of straight Aux lines by individualpinning centers at low temperatures to the new regime ofcollective pinning arising from decoupled quasi-2D pan-cake vortices, taking advantage of pinning by neighbor-ing pinning centers in each plane. In this case, because ofa weak coupling between superconducting planes, theAux line becomes more Aexible to be pinned by many pin-ning centers in different decoupled CuO2 planes, althoughthese centers do not form a straight line parallel to H.The crossover from individual to collective pinning leadsto an enhancement of j, which is evidenced by the de-crease of the dj, /dT slope around T" [see Figs. 10 and11(a)]. A similar j, enhancement caused by the change ofthe pinning mechanism is also observed when the temper-ature is held fixed and instead the applied field isvaried (Fig. 12). From this point of view, the field-
To get an idea to what extent the temperature varia-tion of critical current is related to the change of the pin-ning potential Uo, we have also studied the relaxation ofthe magnetic moment, thus deriving from these data theinformation about the Uo vs T dependences. Flux-creepprocesses were studied in the regime of the remanentmagnetization measured after applying a magnetic fieldH=20 kG, which is larger than the characteristic fieldH* (Ref. 35) of the onset of the full field penetration forboth orientations of H.
The time dependences of P follow a logarithmic de-cay law, at least in the interval t =10 —4X10 s used inour experiments. To separate the temperature variationof Uo from the j(T) dependence, it is important to usethe normalized relaxation rate S =(1/Mo)dM/d(lnt),where Mo is the initial magnetization value. The 5 vs Tcurve has a X-like shape with the maximum atT=T =12 K [Fig. 13(b)]. We should note here thatthe unusual X shape of the S ( T) dependence is now wide-ly discussed in the literature. To compare the tempera-ture variation of the normalized relaxation rate S( T) indifferent oxide superconductors, we have plotted in Fig.11 the j,(T) and S(T) data for Ba, „E„Bi03(BKBO),YBa~Cu30, (YBCO), ' BizSrzCaCuzO(BSCCO), and PSYCCO. In isotropic cubic BKBO,superconductors j ( T) and S ( T) are monotonous func-tions of temperature, with the smaller j,(T) values athigh temperatures [Fig. 11(a)] being in a correspondencewith the larger S(T)—1/Uo(T) values [see Eq. (6) andFig. 11(b)]. In layered YBCO, BSCCO, and PSYCCO su-perconductors, the situation concerning S(T) and j,(T)is more complicated. In quasi-2D BSCCO and also inT128azCa2Cu30 (Ref. 58), there is a kink in the j,(T)dependence at T = T " (Ref. 59) see Fig. 11(a), which isaccompanied by a sharp suppression of the S( T) valuesat the same temperature T=T; see Fig. 11(b).In YBCO superconductors there is a wide plateau on theS ( T) curve [Fig. 11(b)], which is related to themodification of the exponential variation of j, in thesame temperature range. The j,( T) and S ( T) depen-dences for PSYCCO (Fig. 11) look like an intermediatecase between quasi-2D BSCCO and strongly 3D-likeYBCO. The plateaulike dependence of S(T) is quiteoften considered"' as evidence for the existence ofthe vortex glass phase.
Usually the magnetization relaxation process is caused
8218 V. V. METLUSHKO et al. 47
by the thermal excitation of a vortex line over a charac-teristic potential barrier UD.
' This process also leads toa decrease of j, (Ref. 62) and the normalized relaxationrate S(T) is determined by
S(T)=- kTU() kT—ln(t /r, ~)
(6)
There are two well-known relations available between thepinning potential UQ and the critical current. In the An-derson model ' we have
U(j)= Up in(j, o/j) .
The simple substitution of U( j) into Eq. (6) shows thatneither Anderson's ' nor Zeldov's65, 66 model can inter-pret successfully the experimentally observed S(T) pla-teau, since both models predict an upward S (T) curva-ture which at most corresponds to the isotropic BKBOcase but not to the S(T) behavior in layered high-T, ox-ides [Fig. 11(b)].
In the framework of the collective pinning ' and vor-tex glass ' models, we have
U()( T)U(j)=
p
j,p(T)
i.o(»=i, oo 1—
U()(T) = Uoo 1—(10)
where p and n are constants. It is possible to obtain aplateaulike S(T) dependence, though the fitting pro-cedure involves quite a large number of fitting parame-ters. We should not be very enthusiastic at this point,however, since our experimental data for PSYCCO con-vincingly demonstrate that with the increasing decou-pling between CuOz superconducting planes, the plateau-like S ( T) dependence is transformed into the X-like S(T)curve [Fig. 11(b)] with a maximum at T=T =10—20K. Moreover, in quasi-2D BSCCO superconductors thismaximum becomes much sharper, with a drop of S ( T) bya factor of 6 —10 (Refs. 44 —46) in a very narrow ( ~ 2 K)temperature interval. The observation of a very sharpdrop of S ( T) at T = T = 15 K in BSCCO made itnecessary to put forward an idea about changing to a newpinning mechanism at T ~ T (Refs. 44 and 46).
Summing up this section, we note that a continuousdecoupling of the superconducting CuO2 planes results ina strong modification of the S(T) dependences with in-creasing temperature from a monotonously growing S ( T)in cubic BKBO (Ref. 54) compounds through intermedi-ate plateaulike (YBCO) and X-like (PSYCCO) S ( T)curves to a sharp low-temperature anomaly in quasi-2DBSCCO superconductors. The proper theoretical
U( j)= Uo(1 —j/j, p),
whereas in the Zeldov model ' a logarithmic relationholds.
description of the whole range of the S(T) behavior inlayered superconductors with a varying degree of decou-pling between superconducting planes is still lacking. Wemay only suggest here that the sharp low-temperatureS ( T) maximum, which seems to be not reproduced by ei-ther vortex glass ' or by collective pinning ' models,appears to be due to an abrupt onset of a different pin-ning regime (Fig. 11). This speculative idea is also sup-ported by the observation of the pronounced kink in the
j,(T) dependence at T=T [Fig. 11(a)]. One of thepossible scenarios for a change of the pinning mechanismwas proposed by Fisher et aI. In layered superconduc-tors at low temperatures and in low fields, the interlayercoupling is playing a more dominant role, resulting in a3D behavior, though strongly anisotropic. With increas-ing temperature, the crossover from 3D vortex lines toquasi-2D pancakelike vortices occurs, and in each CuOzsuperconducting plane the pancakelike vortices can takeadvantage of being decoupled from other vortices tomove to available neighboring pinning centers to optim-ize their pinning potential. In this case, the full vortexline, consisting of all pancakelike independent vorticessitting one underneath the other in different supercon-ducting CuOz planes, is sufficiently flexible to realize ahigher total effective pinning potential U0 and, conse-quently, a smaller S value. Such a crossover should beaccompanied by the appearance of a power-law behaviorof the current-voltage (I V) charact-eristics.
Current-voltage characteristics can be successfullyused for studies of the dissipation processes in type II su-perconductors. Within the framework of the Kim-Anderson model, ' the flux creep leads to a finite dissipa-tion corresponding to the existence of the finite resis-tance:
VZ~ = lim —,
I~O Iwhich is a linear function of temperature T with the zeroresistance state being realized only at T=0. Recentlynew possible flux phases have been proposed by Fisheret al. ,
' including the vortex glass (VG) phase with thefrozen positions of the individual flux lines. These re-strictions for the flux motion are also accompanied by aweaker dissipation which results in a power law for thecurrent-voltage characteristics for D-dimensional sam-ples 59, 60
(J T —T )—j ( Z + ) ) I( D —) ) (1 1)
where j is the current density, E is the electric field, andT is the phase-transition temperature. For T & T, ac-cording to the VG theory, a negative curvature of the I-Vcurves should be seen on the logI-log V plot, in contrast tothe positive curvature expected from the Anderson-Kimfiux-creep model with V~ sinh(I/Io). ' From this pointof view, the change of curvature of the logI-logV plotmay be interpreted as an indication of the transition intothe vortex phase.
Conventional methods of the I-V measurements arelimited by poor voltage resolution. On the other hand,the derivation of the voltage from the flux-creepdata has an advantage of making available the pico-
LOWER CRITICAL FIELDS, CRITICAL CURRENTS, AND. . . 8219
1 dN2mR dt
3poI dP
dt(12)
where P and R are the magnetic moment and the radiusof the sample, respectively, and I =3.328 (Refs. 73,74).
As in classical type II superconductors, e.g. , Nb-Ti andNb3Sn, where V=kI", with n =15—150 (Ref. 75), wehave also found quite large n values for the PSYCCO sin-gle crystals: n -25 —45. All our flux-creep measurementshave been done in the regime of the critical state, i.e.,when the applied magnetic field H =2T was much higherthan the field H* of the onset of the full field penetration.The I-Vcurves (Fig. 14) follow the power law
& =&o(jljo)", (13)
with the exponent n being strongly suppressed by mag-netic field. We have also tried to fit an exponential rela-tion between j and E as predicted in Ref. 61, but underthese conditions the fitting gets worse. For long enoughtimes (t ~ 100 s) the n value in the power law [Eq. (13)] isinversely proportional to the normalized relaxationrate:
E(Vim)
10
10
10
30K 25K
i
20K 15K10K T~5K
12.5K, 7 5K]j I ] a
0 e
volt range for current-voltage characteristic studies. Theunderlying idea is quite simple. The shielding current it-self can easily be found from the corresponding width ofthe magnetic-moment hysteresis loop, while the electricfield E may be obtained on the basis of the Faraday in-duction law
d( lnE) d( lnP )n=d(lnj) d( lnt)
(14)
Therefore the exponent n in Eq. (13) is expected to have aminimum at temperatures where the S(T) dependencehas a maximum (Fig. 15). To check the validity of the re-lation nS =1 [see Eq. (14)], we have plotted the productnS vs T (Fig. 16), with both parameters determined fromthe magnetization measurements: S as the normalizedflux-creep rate and n as the slope of the lnE vs lnj plots(Fig. 14). It turns out that the simple relation nS =1,originating from the Faraday law if the E vs j depen-dence really corresponds to Eq. (13), is fulfilled in thewhole range of temperatures T ~ 60 K used in our experi-ments [see Figs. 16(a) and 16(b)]. The typical n values arein the range 25 —45 (Fig. 15), which implies the presenceof a quite sharp, well-defined critical behavior for the I-Vcurve on a picouolt voltage scale (Fig. 14). The latter is avery low value not so easily achieved in conventionalmethods of I-V measurements.
Next we would like to turn to the discussion of the an-isotropy e6'ects. First of all, we are starting from a re-markable similarity between the S(T) dependences forboth H~~c and Hlc orientations [Fig. 13(b)]. There aresome reasonable arguments given above for the interpre-tation of the appearance of the S(T) maximum at lowtemperatures for the H~~c orientation. These arguments,however, seem to be quite irrelevant to explain the sameS(T) behavior for the Hj.c orientation. We think thatthe similarity of the S(T) behavior for H~~c and Hlc [Fig.13(b)] is directly caused by a small (a- I' —3') misorienta-tion of H with respect to the exact Hlc geometry. Such amisorientation, which seems to be quite realistic in acryostate without built-in facilities to rotate the sample inthe course of measurements at low temperatures, is negli-gible for weakly anisotropic systems, but may be of vitalimportance for strongly anisotropic compounds. As ithas been shown in Refs. 29—31, at nearly all angles, ex-
10-1050
20000I
200000j{A/cm 40-
E(vim)108
1060K
55K
35K50K 45K 4OK
30-
&o o020-
a (-j aQ 0
1O"= 10- oHJc
-114000 2
j{A/cm )4000 0 I I I I I I
0 10 20 30 40 50 60 70 80T{K}
FIG. 14. Current-voltage characteristics of aPb2Sr2Y, „Ca Cu308+y single crystal at 8=0.
FICx. 15. Temperature dependence of the exponent n in Eq.(13) for PbzSr2Y, „Ca Cu308+~, H=0: =H~~c, E =Hjc.
8220 V. V. METI.USHKO et aI. 47
S n
2[Hlt c
1.5
Cl~Q&gLI 0 & + Cl g Q p Cl
0.5I-
I
QL ——
(a)
0 10 20 30 40 50 60 70T(K)
S n
2iHJc
1.5—
oCX3 0 o o o 0 0
0
0.5—
(b)
0 f t
0 10 20 30 40 50 60 70 80T(K)
phase transition or a crossover at H =H -4 kG(T =30 K) and at H=H -2 kG (T =50 K) (see Fig.18), where there is a distinct S-shaped anomaly on theI-V curve (Fig. 17). As a result, a kink at H=H isseen in the n vs H dependence (Fig. 18). For T =30 Kthe S-shaped anomaly on the I-V dependences, implyingthe change of the curvature, is clearly observed twice (seeFig. 17) at H=4 kG and H=8 kG. A similar change ofthe curvature has been reported for Bi2Sr2CaCu20~ inRef. 77. Following Fisher et aI. ,
' we propose below atentative explanation of the variation of the vortex phasesas a function of applied field. Firstly, for very strong an-isotropy, when
FIG. 16. Sn vs T for Pb&Sr2 Y& „Ca„Cu308+„, H =0:
=Hiic, o =Hie.L, a, (4O/B )'/
L,b d d(17)
cept a-+1'—3' from the exact Hlc configuration, themagnetization and its dynamics are determined only bythe magnetization component M ' perpendicular to theCu02 layers, which is much larger than the parallel com-ponent M ' (Ref. 32):
where a, =—(40/B)' is a lattice constant, the wavelengthof the dominant Auctuations becomes smaller than theinterplanar separation d; the system has essentially a 2Dcharacter. This is shown in the H-T phase diagram inFig. 19 above the nearly horizontal dashed line, corre-
Mlc
16mLbp(.8, )
B~'P(O )
@tan(O, )
16vr L bI P(O, )
X ln
c2
Bll P(O)
I /21——+a3D
I /21 +a3D
(15)
(16)
where P(O, ) = [1+1 tan (0, )]'/, a3D=O. 5, and g= l.These are expressions for the reversible magnetization
in quasi-2D layered superconductors. The ratioM '/M ' is given by the anisotropy I and the angle 0,between the vortex lattice and the c axis:
Mll& I-2
tan(O, )
E(V/m}
10 8
(~)
10
ee~ ~
~ ~
~ ~~ ~
~ ~~ ~
e
~ ~~ ~
ee
~ ~
~ ~
4000G& ~
'
—1010
—3000625006
ee
ee
e
e
e
160
~ ~~ ~
~ ~
ee
~ ~~ ~
~ ~
~ ~
~ ~~ e
e e~ ~
1100G
1200G
1120000 2
j(A/cm )
E(V/m}
10
2000G 1400G
e
~ ~~ ~
~ ~
T 30K
40000
For I -31, the difference M~~'=10 M ' may be ex-plained by the presence of a very small misorientation an-gle a=(~/2) —8, —1. From this point of view, thestriking similarity between S "'( T) and S '( T) [Fig.13(b)] may be easily interpreted as a result of this uncon-trollable misorientation, because for tilted fields aroundHlc, the S '(T) dependence is, in fact, determined bythe presence of the small M"'(H) component and there-fore both SH '(T) and S "'(T) curves should have thesame shape, as it is found experimentally [Fig. 13(b)].Taking this misorientation problem into account, we areconcentrating our efforts only on the discussion of theH~~c orientation.
The same crossover in the pinning regime may alsooccur if one keeps the temperature constant but variesthe magnetic field instead. For the fixed temperatures( T =30 K, Fig. 17) we have found a clear indication of a
109 =
—1010
(b)
200006e
15000G~12000
ree
e
ee l' e J
~ ~
~ efe~ ~ ~
~ eo~ ~ ~
~ ~~ ~ ~e e e ~
~ ~ e ~~ I ~
~ ~ 0 ~~ ~ l ~
~ e e ~~ ~ g ~
~ ~ ~~ ~ e ~
~ eee e~ ~ ee ~
~ ~ ~ e ~~ ~~ ~ e ~
e
5000G~ ~
6000G
7000GOG
6000 2j(A/cm }
26000
FIG. 17. Current-voltage characteristics of aPb2Sr2Y& Ca Cu308+y single crystal at T =30 K for differentmagnetic fields.
47 LOWER CRITICAL FIELDS, CRITICAL CURRENTS, AND . ~ ~ 8221
30(
25
20
15
10
00 5 10 15 20 25 30 35
H(kG)
sponding to L /L =a d,b —,/d. For this 2D regime the melt-ing line is approximately the same as the Kosterlitz-Thouless-type melting transition T '
2Dogarithmically interacting point vortic hices i ac u co-
NdTM =(1—2) X10
16m L,b
0
=(20—40 K) L.b(TM ) 10 A
=7—14 K. (18)
In PSYCCO, with L,b =1860 A and d =12.3 A the
FIG. 18. Field dield dependence of the exponent n in Eq. (13) forPb&Sr& Y~, Ca„Cu&O&+~ at = T =30 K, Q' = T =50 K.
crossover line I /L =afield of 10 kG Th
,b—a„corresponds to a magnetic
kG. The TM value, estimated for PSYCCOfrom Eq. (18), lie in the range 7 —14 K. Below TM inmagnetic fields H) H a 2D vortex 1 2Dvor ex g ass or 2D vortexlattice state should be realized th f' e, e ormer being favoredby the presence of disorder. For H &H ht e supercon-
uc ing ayers are coupled and the 3D anisotropic behav-ior takes place with the melting temperature:
1/2
T3D 2 ab +0M CL (19)
Q2
16~2Lab
where cL =0.15 is a constant. Thus, for H&H thevs (see Fig. 19).melting line follows the T v (H)
Below the melting line a 3D vortex lass (or 3Dlattice stateate is reahzed, depending upon the strength ofthe disorder potential present in the system.
Now, coming back to our I Vdata (F-ig. 17), we maygive the following interpretation. Than 0 K used for the I- V measurements are higher than
wo c aracteristice s or which an S-shaped anomaly is seen in the I-V
curves; for T=30 K, for example, at H=4 kG and H=8ig. &. he first field seems to correspond to the
crossing of the melting line T [E . (19), hq. , w ereas thesecond field may be related to the crossover field H(horizontal dashed line in Fig. 19) d'ig. , corresponding to the
reduction of the dimensionality. The value ofH is temperature independent if the temperature vtion of L 'E[ q. (18)j is not taken into account. The transi-
pera ure vana-
tion into the 2D vortexb th
iquid phase is also accompaniedy the change of the slope of the j, vs H (Fig. 12) and S
vs H curves (not shown).~ ~
This interpretation of the appearance of the two S-shaped anomalies on the I-V plot
'1p o s is on y a qualitative
tentative picture, which should be further elaborated on aquantitative level. Of course it do tc u e the possible validity of other alternative explana-tions.
V. CONCLUSIONS
2Dvortexglass
orvortexlat tice By
3Dvortex gla ovortex lattice
Meissner phase
FIG. 19. Schematic phase diagram of a type II la er dconductor aor as a function of applied magnetic field and tempera-ture.
We have studied temperature dependences of lowercritical fields, critical currents d th 6, an e ux-creep rate onnear y cube-shaped 0.6X0.6X0.5 mm PSYCCO
y wo different field orientations, H~~c and Hlc.Lower critical fiel il'elds H„(T) are characterized by ananomalous curvature at low temperatures, which may beinterpreted in terms of models h' h kt"e'a ered S-N-
w ic ta e into accounth 1 y ( -N-S or other types) structure of these oxide
superconductors with proxim't -' d d~ ~ ~
i y-in uce superconduc-tivity in the layers separating th C Oe u 2 superconductingplanes (Refs. 22 —28 and 78,79). The temperature depen-dences of the critical current jH"'(T) d h dAux-creep rate S ' T i
an t e normalized( ) indicate that with increasing tem-
perature a crossover from the anisotropic 3D to uasi-2D'
r a es place, leading to a strong modification ofthe pinning mechanism. In comparison with oth h' h-
'des, the amsotropy I"—:(m /m )' —31mab is 0 anmtermediate value between YBCO ( I = 5 )
(1 =50-140'. u&. Such a large anisotropy parameter implies
8222 V. V. METLUSHKO et al. 47
that in order to be sure that the magnetic field H is reallyparallel to the CuOz planes, it is necessary to rotate thesample in situ in the magnetometer by very small anglesabout the Hlc orientation. Unfortunately we could notdo that in our measurements and, as a result, all datagiven here for Hlc are only within +3 from the exactHic orientation. This small miso rientation may besufficient to have quite a strong influence of the magneti-zation component perpendicular to the Cu02 planes. Be-cause of these reasons the values of the anisotropies ofthe lower critical fields and critical current given aboveshould be considered only as a lower limit and the realanisotropy might even be higher.
Using current-voltage characteristics in the picovoltrange, which were derived from the damping electric field
determined by the magnetic-Aux-creep rate, we havefound that the decoupling of the pancakelike vortices indifferent Cu02 planes, occurring at T=T, leads notonly to the j,(T) and S (T) anomalies, but also to thechange of the exponent n in the expression [Eq. (13)] re-lating j to the electric field E.
ACKNOWLEDGMENTS
We thank A. A. Zhukov and V. D. Kuznetsov forhelpful discussions. V. Metlushko would like to acknowl-edge the support by DAAD and DFG SFB 341. V. V.Moshchalkov has benefited from the financial support bythe Research Council of the Katholieke Universiteit Leu-ven.
*Also at Laboratory of High- T, Superconductivity, PhysicsDepartment, Moscow State University, 117234, Moscow,Russia.
iR. J. Cava, B. Batlogg, J. J. Krajewski, L. W. Rupp, L. F.Schneemeyer, T. Siegrist, R. B. van Dover, P. Marsh, W. F.Pech, Jr., P. K. Gallagher, S. H. Glarum, J. H. Marshall, R.C. Farrow, J. V. Waszczak, R. Hull, and P. Trevor, Nature(London) 336, 211 (1988).
2M. M. Lukina, V, N. Milov, and V. V. Moshchalkov, Sov.Phys. Supercond. 3, 2767 (1990).
J. S. Xue, M. Reedyk, Y. P. Lin, C. V. Stager, and J. E.Greedan, Physica C 166, 29 (1990).
4M. Reedyk, C. V. Stager, T. Timusk, J. S. Xue, and J. E.Greedan, Phys. Rev. B 44, 4539 (1991).
~C. Chaillout, O. Chmaissen, J. J. Capponi, T. Fournier, G. J.McIntyre, and M. Marezio, Physica C 175, 293 (1991).
A. K. Pradhan, S. B. Roy, Chen Changkang, J. W. Hodby, andD. Caplin, Physica C 197, 22 (1992).
7Magnetic Property Measurement System, ApplicationNotes/Technical Aduisories (Quantum Design, San Diego,1990), p. 110.
sL. D. Landau and E. M. Lifshits, Electrodynamics of Continu-ous Media (Pergamon, London, 1960), p. 417.
A. A. Gippius, V. V. Moshchalkov, and V. V. Pozigun, Fiz.Tverd. Tela (Leningrad) 30, 765 (1988) [Sov. Phys. Solid State30, 438 (1988)].M. Wacenovsky, H. W. Weber, O. B.Hyun, D. K. Finnemore,and K. Mereiter, Physica C 160, 55 (1989).A. P. Malozemoff, L. Krusin-Elbaum, D. C. Cronemeyer, Y.Yeshurun, and F. Holtzberg, Phys. Rev. B 38, 6490 (1988).L. Krusin-Elbaum, R. L. Greene, F. Holtzberg, A. P.Malozemoff, and Y. Yeshurun, Phys. Rev. Lett. 62, 217(1989).
' V. V. Moshchalkov, A. A. Zhukov, V. D. Kuznetsov, V. V.Metlushko, V. I. Voronkova, and V. K. Yanovskii, SolidState Commun. 74, 1295 (1990).
' V. V. Moshchalkov, J. Y. Henry, C. Marin, J. Rossat-Mignod,and J. F. Jacquot, Physica C 175, 407 (1991).
' M. Naito, A. Matsuda, K. Kitazawa, S. Kambe, I. Tanaka,and H. Kojima, Phys. Rev. B 41, 4823 (1990).D. E. Farrell, S. Bonham, J. Foster, Y. C. Chang, P. Z. Jiang,K. G. Vandervoort, D. J. Lam, and V. G. Kogan, Phys. Rev.Lett. 63, 782 (1989).R. B. van Dover, L. F. Schneemeyer, E. M. Gyorgy, and J. V.Waszczak, Phys. Rev. B 39, 4800 (1989).
L. A. Gurevich, I. V. Grigor'eva, N. N. Kolesnikov, M. P.Kulakov, V. A. Larkin, and L. Ya. Vinnikov, Physica C 195,323 (1992).D. E. Farrell, C. M. Williams, S. A. Wolf, N. P. Bansal, andV. G. Kogan, Phys. Rev. Lett. 61, 2805 (1988).V. Welp, W. K. Kwok, G. W. Crabtree, K. G. Vandervoort,and J. Z. Liu, Phys. Rev. Lett. 62, 1908 (1989).G. J. Dolan, F. Holtzberg, C. Field, and T. Dinger, Phys. Rev.Lett. 32, 2184 {1989).M. Tachiki, S. Takahasi, F. Steglich, and H. Adrian, Z. Phys.B Condens. Mater. 80, 161 (1980).L. N. Bulaevskii, Int. J. Mod. Phys. B 4, 1849 (1990).
24A. I. Buzdin, V. P. Damjanovic, and A. Yu. Simonov, Phys.Rev. B 45, 7499 (1992).A. A. Abrikosov, Physica C 182, 191 (1991).T. Koyama, N. Takezava, and M. Tachiki, Physica C 168, 69(1990).
V. M. Krasnov, Physica C 190, 357 (1992).2~A. A. Golubov and V. M. Krasnov, Physica C 196, 177 {1992);
V. M. Krasnov, V. A. Oboznov, and V. V. Ryazanov, ibid.196, 335 (1992).
A. V. Balatskii, L. I. Burlachkov, and L. P. Gorkov, Zh. Eksp.Tear. Fiz. 90, 1478 (1986) [Sov. Phys. JETP 63, 866 (1986)].A. I. Buzdin and A. Yu. Simonov, Physica C 175, 143 (1991).
iA. I. Buzdin and A. Yu. Simonov, Physica B 165-166, 110(1990).D. Feinberg, Physica C 194, 126 (1992).A. M. Campbell and J. E. Evetts, Adv. Phys. 21, 199 (1972).C. P. Bean, Phys. Rev. Lett. 8, 250 (1962).C. P. Bean, Rev. Mod. Phys. 36, 31 (1964).V. V. Moshchalkov, A. A. Zhukov, L. I. Leoniuk, V. D.Kuznetsov, and V. V. Metlushko, Sov. Phys. Supercond. 2, 84(1989).V. G. Kogan, M. M. Fang, and S. Mitra, Phys. Rev. B 38,11 958 (1988).
E. W. Scheidt, C. Hucho, K. Luders, and V. Muller, SolidState Commun. 71, 505 (1989).B. Pumpin, H. Keller, W. Kundig, W. Odermatt, I. M. Sovic,J. W. Schneider, H. Simmler, P. Zimmermann, E. Kaldis, S.Rusiecki, Y. Maeno, and C. Rossel, Phys. Rev. B 42, 8019(1990).S. Mitra, J. H. Cho, W. C. Lee, D. C. Johnston, and V. G. Ko-gan, Phys. Rev. B 40, 2674 (1989).J. R. Thompson, Yang Ren Sun, L. Civale, A. P. Malozemoff,M. W. McElfresh, A. D. Marwick, and F. Holtzberg (unpub-
47 LOWER CRITICAL FIELDS, CRITICAL CURRENTS, AND. . . 8223
lished).4~A. P. Malozemoff, in Physical Properties of High Temperature
Superconductors, edited by D. Ginsberg (World Scientific,Singapore, 1989), p. 71 ~
~sW. J. Yen, Z. Q. Yu, S. Labroo, and J. Y. Park, Physica C194, 141 (1992).
44A. A. Zhukov, V. V. Moshchalkov, V. A. Rybachuk, V. A.Murashov, A. Yu. Martynkin, S. W. Moshkin, and I. N. Gon-charov, Physica C 185-189, 2137 (1991).
4~V. N. Zavaritsky and N. A. Zavaritsky, Physica C 185-189,2141(1991).A. A. Zhukov, V. V. Moshchalkov, V. A. Rybachuk, V. A.Murashov, and I. N. Goncharov, Zh. Eksp. Teor. Fiz. 53, 466(1991) [Sov. Phys. JETP Lett. 53, 489 (1992)].
~7D. Neerinck, K. Temst, M. Baert, E. Osquiguil, C. VanHaesendonck, Y. Bruynseraede, A. Gilabert, and I. A. Schull-er, Phys. Rev. Lett. 67, 2577 (1991).
4~M. Daeumling, J. M. Seuntjens, and C. Larbalestier, Nature(London) 256, 332 (1990).S. Senoussi, M. Oussena, G. Collin, and I. A. Campbell, Phys.Rev. B 37, 9792 (1988).
~OC. N. Kopylov, A. E. Koshelev, I. F. Schegolev, and T. G. To-gonidze, Physica C 170, 291 (1990).N. Chikumoto, M. Konczykowski, N. Motohira, K. Kishio,and K. Kitazawa, Physica C 185-189, 2201 (1991).
~~V. V. Moshchalkov, A. A. Zhukov, G. T. Karapetrov, V. D.Kusnetov, V. V. Metlushko, V. I ~ Voronkova, and V. K.Yanovskii, Physica B 169, 653 (1991).K. Kadowaki and T. Moshiku, Physica C 195, 127 (1992).
~4G. T. Seidler, T. F. Rosenbaum, P. D. Han, D. A. Payne, andB.W. Veal, Physica C 195, 373 (1992).A. P. Malozemoff and M. P. A. Fisher, Phys. Rev. B 42, 6784(1990).Y. Xu, M. Suenaga, A. R. Moodenbaugh, and D. O. Welch,Phys. Rev. B 40, 10 882 (1989).
~7I. C. Campbell, L. Fruchter, and R. Cabanel, Phys. Rev. B 64,1561(1990)~
~~M. Mittag, M. Rosenberg, A. V. Niculescu, B. Himmerich,and H. Sabrawsky, Physica C 198, 222 (1992).
~9D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev. B
43, 130 (1991).M. P. A. Fisher, Phys. Rev. Lett. 62, 1415 (1989).P. W. Anderson and Y. B. Kim, Rev. Mod. Phys. 36, 39(1964).J. R. Thompson, Y. R. Sun, and F. H. Holtzberg, Phys. Rev.B 44, 58 (1991).C. W. Hagen, R. P. Griessen, and E. Salomons, Physica C 157,199 (1989).C. W. Hagen and R. P. Griessen, Phys. Rev. Lett. 62, 2857(1989).
6~E. Zeldov, N. M. Amer, G. Koren, A. Gupta, M. W. McEl-fresh, and R. J. Gambino, Appl. Phys. Lett. 56, 680 (1990).E. Zeldov, N. M. Amer, G. Koren, and A. Gupta, Appl. Phys.Lett. 56, 1700 (1990).M. V. Feigel'man, V. B. Geshkenbein, A. I. Larkin, and V. M.Vinokur, Phys. Rev. Lett. 63, 2303 (1989).
~M. V. Feigel'man and V. M. Vinokur, Phys. Rev. B 41, 8986(1990).R. H. Koch, V. Foglietti, W. J. Gallagher, G. Koren, A. Gup-ta, and M. P. A. Fisher, Phys. Rev. Lett. 63, 1511 (1989).G. Ries, H. W. Neumuller, and W. Schmidt, Supercond. Sci.Technol. 5, S81 (1992).W. Paul, D. Hu, and Th. Bauman, Physica C 185-189, 2373(1991).J. E. Tkaczyk, R. H. Arendt, M. F. Garbauskas, H. R. Hart,K. W. Lay, and F. E. Luborsky, Phys. Rev. B 45, 12506(1992).A. A. Zhukov, Solid State Commun. (to be published).
74V. V. Metlushko, A. A. Zhukov, V. V. Moshchalkov, V. D.Kuznetsov, M. M. Lukina, and V. N. Milov (to be published).J. E. Evetts and B.A. Glowacki, Cryogenics 28, 641 (1988).
76S. Senoussi, J. Phys. (Paris) (to be published).Y. Ando, N. Motohira, K. Kitazawa, J.-I. Takeya, and S. Aki-ta, Phys. Rev. Lett. 67, 2737 (1991).A. I. Buzdin, A. Yu. Simonov, and V. P. Damjanovic, Zh.Eksp. Teor. Fiz. 53, 503 (1991) [Sov. Phys. JETP Lett. 53, 528(1991)].L. N. Bulaevskii and M. V. Zyskin, Phys. Rev. B 42, 10230(1990).
