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Magnetic relaxation in high-temperature superconductors
10 May 1999 PHYSICS
LETTERS A
Physics Letters A 255 (1999) 183-186
Magnetic relaxation in high-temperature superconductors Hao Jin, Lin Chen, Yu Heng Zhang Structure Research Laboratory, University of Science and Technology of China, Hefei 230026, P.R. China
Received 18 November 1998; accepted for publication 17 February 1999 Communicated by J. Flouquet
Abstract The magnetic relaxation rate was calculated for the vortex-glass or collective-creep model and the thermal activated model considering backward hopping. By comparing the theoretical results with experimental data it was found that the former models were in good agreement with the experiment at relatively low temperature regions, while the latter were at higher temperatures near T,. These results are discussed in detail. @ 1999 Elsevier Science B.V. PACS: 74.fN.Ge Keywords: relaxation rate; Flux creep; Flux dynamics; Thermal activated; Pinning; High-temperature superconductors
1. Introduction
The flux dynamics of high-temperature superconductors (HTSC) are particularly different from those of conventional superconductors. One of those important phenomena found in experiments is the non-linear relationship between M and In t, which does not fit the Kim-Anderson model [ 1,2]. Various models were proposed for explanations, including the vortex glass model by Fisher et al. [3,4], the collective-creep model by Fiegle’man et al. [ 561, the logarithmic j-dependent barrier model by Zeldov et al. [ 71, the revised Kim-Anderson model by Hagen and Griessen [ 81, and the creep model with damp by Yin et al. [ 91. All the theoretical results of these models, to some extent, can give an explanation of the experiments. In the interest of finding out more about the validity of each model, the relaxation rate s = -dM/d In t and the normalized relaxation rate = s/M were introduced so s, = -(l/M)dM/dlnt that the magnetic relaxation behavior can be described more sensitively. The experiment [ lo] showed that there was a plateau for sn in the s,-T curve from 20 K to 60 K. Theoretically, the vortex-glass or collectivecreep model gives a plateau in the s,-T relationship, which agrees with experiments. However, the logarithmic barrier model can not derive this result, and
the Kim-Anderson model gives a negative plateau result at long times and high temperatures, which does not agree with the experiments [ 111. On the other hand, experiments showed that the snT relationship exhibits a rising trend [ lo], or in the s-T relationship [ 121, s drops rapidly as T increases near T,. The reason of s,-T rising is that the dropping speed of M with T is much bigger than that of s with T. Theoretically, explanations in this temperature region have not been reported although these models have been studied for years. The relaxation rate s, rather than the normalized relaxation rate s,, is discussed in this paper while considering the following two points: first, in the vortexglass model or the collective-creep model, there is an inverse ratio relationship between sn and the parameter I(, while the s-p relationship obeys a power law and is thus more sensitive: secondly, both M and s drop as T rises, so that s, = s/M is a variable which changes slowly with T. In the present work, the relaxation rate s was derived from the vortex-glass model or the collective-creep model and compared with the experiment [ 121. Moreover, considering the relatively low activation barrier Ue in HTSC and the rather high working temperature T near T,, the contribution of backward hopping [ 13,141, which is the hopping of flux lines from low
0375-9601/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved.
184
H. Jin et al./Physics
Letters A 255 (1999) 183-186
barriers to high ones, was introduced into the thermal activated model, and the derived s was also compared with the experiment. The results suggest that the vortex-glass or the collective-creep model will dominate the flux behavior at relatively low temperatures, but as the temperature increases, especially when T is near T,, the thermal activation with both forward and backward hopping will play a more important role in the flux-relaxation process. Discussion of this are given in this paper.
2. Calculated s-T dependence 2.1. Vortex-glass model and collective-creep model
In both the vortex-glass model [3,4] and the collective-creep model [5,6] the pinning barrier satisfies the equation U(j)
2.2. Thermal activated model considering backward hopping
In the thermal activated model, while considering backward hopping, the hopping rate Y satisfies y=Zexp(
In Eq. (6), j represents the current density, B represents the average magnetic induction, d represents the size of pinning center, and vo represents the hopping rate of the initial state at which the relaxation does not happen. When t = to (Y = vg), namely at the beginning of relaxation, we have dB/dx = pajc. Also, noticing that there is no flux in the pinning center at that state, which means a forward hopping only, one obtains l=exp(
K=Zexp( where j is the electric current density, j, is the critical electric current density. The parameter ,u is less than one [ 31 for the vortex-glass model. In the collectivecreep model [ 61, p varies among values l/7, 312, 719, 918, l/2, depending on different temperatures and fields. Obviously the barrier U(j) diverges as j goes to zero. From Eq. ( 1) , using the Arrhenius relationship
t
VO
exp
U(j)
[
- -
kT
1
(2)
we have
(3) Defining the relaxation rate dM ‘= -dint
S=--dIn-=
-g)exp($)
(7)
-s)sinh($).
UO
(8)
Comparing Eq. (8) with Eq. (2)) we have U(j)=Ue-kTln(2sinh($$)).
(9)
Thus, when j ---)0, U(j)
= UO+ kTln
j&T 2juo’ ( )
(10)
It is obvious that U(j) diverges when j --t 0. Noting that j + 0 means that the Lorentz force approaches zero, so the forward and backward hopping rates become the same. These correspond to an infinite effective barrier U(j) in the Arrhenius’ equation (2). Thus the divergence of U(j) as j approaches zero has a clear physical meaning physics. Using relation Y/VIJ= to/t in Eq. (8) and solving the algebraic equation leads to
(4)
and substituting (3) into (4), one obtains dM
(6)
Combining Eqs. (6) and (7), we find
=UO[($)p11.
to y -_=-_=
-$)sinh(g).
VO
+ln[~+~/l+4~exp(-2/$)]
d(j/j=) d In t
!Jl+ $1, (-f-)I-“““.
(11)
Substituting Eq. ( 11) into Eq. (4)) we find
(5)
sfx-
kT Uo Jl
1 + 4( t/to)* exp( -2Uo/kT)
(12)
185
H. Jin et al./Physics Letters A 255 (1999) 183-186
3.0 1
U,=38.8meV
3.0
t/t,=3.8x103
lJ,=38,8meV
1/1,=3.8x103
t 2.5
0.0 0
20
40
60
80
100
T WI
T(K) Fig. 1. Calculated s-T relationship from Eq. (4) with different p. according to the vortex-glass or collective-creep model (solid lines). The stars represent the experimental data. Calculating parameters are Un = 38.8 meV, t/to = 3.8 x 103.
Fig. 2. Calculated s-T relationship from Eq. (9). according to the thermal activated model considering the backward hopping (solid line). The stars represent the experimental data. Calculating parameters are Un = 38.8 meV, t/to = 3.8 x 103.
3. Comparison of the theoretical results with the experimental data
well at high temperatures and s will go to zero as T goes to TC. However, the fit in the low temperature range is not good. The behavior of s at temperatures near T, as mentioned above is reasonable. From the physical point of view, the superconductive state goes back to a normal state when T = T, and M = 0, thus s = 0 necessarily. For type-II superconductors, more magnetic flux will penetrate the superconductors when the temperature increases, thus the proportion of the normal state part will increase, and M decreases. When T + T,,
3.1. Vortex-glass model and collective-creep model The relaxation rate s, derived from either the vortexglass model or the collective-creep model, satisfies Eq. (5), with the only difference being in the value of the parameter ,u, so s is discussed at the same time. Fig. 1 shows the calculated s-T relationship according to Eq. (5) with different ,X values. The experimental data in Fig. 1 were obtained from the work of Yeshurun, Malozemoff, Holtzberg et al. [ 121, which were measured on a YBCO single crystal at an applied field of 1 kOe. It is obvious that the temperature corresponding to the maximum s moves to the lowtemperature side as p decreases. The theories showed good agreement with the experiment at low temperatures when ,U is between l/2 and 1. But it was also noted that at high temperatures the theoretical results did not fit the experiment at all. In fact, it is obvious that s never approaches zero as T goes to T, with p ranging from l/2 to 1. When p is smaller, although the calculated s (see the ,X = l/7 curve) will reach zero at high temperatures, the fitting with experimental data is very poor over the full temperature range.
3.2. Thermal activated model considering backward hopping
The s-T relationship in Eq. ( 12) is shown in Fig. 2. As can be seen, the calculated results fit the experiment
M + 0.
Therefore it is concluded from the comparison of those models with the experiment (see Fig. 3, with parameter ,X = 0.65) that the vortex-glass model and the collective-creep model can give a good explanation of s at relatively low temperatures, while the thermal-activated model considering backward hopping can give a good explanation of s at relatively high temperatures.
4. Discussion It appears from the results mentioned above that to describe the relaxation rate s correctly in HTSC, both the vortex-glass or collective-creep model and the thermal activated model considering backward hopping should be used at different temperature regions. That is to say, there exists a cross-over from collected barriers to a single barrier as temperature rises. This conclusion is reasonable from a physical
H. Jin et al./Physics
186
3.0
Letters A 255 (1999)
.,*--*I, t
2.5
vortex
-_
/
p=odi
2.0
/’
1.5 v)
,‘I
I
1.0 -: 0.5
,I
.: _: :”
o.of 0
20
40
60
80
100
T(K) Fig. 3. Comparison between theoretical results and experimental data. The dotted line represents the vortex-glass model, the solid line represents the thermal activated model considering the hack-
ward hopping, the stars represent the experimentaldata. Calculat-
183-186
activation barrier Ua and the higher temperature (near T,) for HTSC, the contribution of flux hopping from low potentials to high ones can not be neglected. Therefore, by adding the backward flux hopping, the thermal activated model will account for the flux creep behavior at a relatively high temperature region and it is in good agreement with the experimental data. The magnetic relaxation in HTSC is a very complex question and is still not perfectly understood. We would like to mention some alternative points of view in recent work, for instance that (elastic) collective creep is valid at low temperatures or field then plastic creep wins over [ 151, or collective creep still holds but with a non-monotonous variation of the exponent ,U [ 161 until the (second-order) melting or irreversibility line is reached. Acknowledgements
ing parameters are Un = 38.8 meV, t/to = 3.8 x 103.
point of view. First, we consider the vortex-glass model to be responsible for the behavior of the magnetic relaxation at relatively low temperatures. When the temperature reaches T,, melting of the vortex lattice will occur and the system will get into the vortex liquid region. However, the pinning still exists. Hence the individual pinning in the Rim-Anderson model, rather than the vortex glass, is suitable for the description of this situation. From the experimental data, however, it can be noted that the plateau region is below the irreversibility line or the melting line. This fact is not difficult to understand, because, in general, there is not a clear boundary of transition from solid to liquid for a glass, i.e. a wide temperature range of transition should exist. Therefore, it is plausible that the cross-over may be related to the premelting softening of the vortex lattice. Secondly, according to the collective-creep model, a bundle of flux lines is pinned by several pinning centers at the low temperature region. The collective-creep model considers the case of identical barriers. In actual cases, the barriers should have some distribution. So with increasing temperature and the enhancement of thermal activation, the weaker pinning centers will be depinned. Thus when the temperature rises high enough, all other pinning centers will have no effect except for the strongest pinning site due to thermal depinning. Because of single-barrier pinning, the elastic or plastic energy then vanishes and the pinning energy will dominate the system, even that cross-over occurs in the vortex solid region [ 121. Moreover, considering the smaller
This work is financially supported by the National Center for Research & Development on Superconductivity and Foundation of Academia Sinica.
References [II Y.B. Kim, C.F. Hempstead, A.R. Stmad, Phys. Rev. Lett. 9 (1962) 306.
I21 PW. Anderson, Phys. Rev. L&t. 9 (1962) 309. [31 M.P.A. Fisher, Phys. Rev. Lett. 62 (1989) 1415. [41 M.P.A. Fisher, D.S. Fisher, D.A. Huse, Physica B 169
(1991) 85. [51 M.V. Fiegle’man, V.B. Geshkenbein, [61 [71 ISI [91 1101
[Ill [I21 [I31 1141
Cl51 [I61
A.l. Larkin, V.M. Vinokur, Phys. Rev. Lett. 63 (1989) 2303. M.V. Fiegle’man, V.B. Geshkenein, V.M. Vinokur, Phys. Rev. B 43 (1991) 6263. E. Zeldov, N.M. Amer. G. Koren et al., Phys. Rev. Lett. 62 (1989) 3093. C.W. Hagen, R. Griessen, Phys. Rev. Lett. 62 (1989) 2857. D. Yin, W. Schauer, V. Windte, H. Kupfer et al., Phys. B 94 (1994) 249. L. Civale, A.D. Marwick, M.W. McElfresh, T.K. Worthington, A.P. Malozemoff, F.H. Holtzberg, J.R. Thompson, M.A. Kirk, Phys. Rev. Lett. 65 ( 1990) 1164. Y. Yeshurun, A.P. Malozemoff, A. Shaulov, Rev. Mod. Phys. 68 (1996) 911. Y. Yeshurun, A&? Malozemoff, Phys. Rev. Lett. 60 (1988) 2202. D. Dew-Hughes, Cryogenics 28 ( 1988) 674. C.W. Hagen, R.P. Griessen, in: Studies of High Temperature Superconductors: Advances in Research and Applications, Vol. 3, Ed. A.V. Narliar (Nova Science, New York, 1990) p. 159. Y. Abulatia, A. Shaulov, Y. Wolfus, R. Prozorov, L. Burlachkov, Y. Yeshurun, Phys. Rev. Lett. 77 ( 1996) 1596. T. Klein, W. Harneit, L. Baril, C. Escribe-Filippini, D. Feinbcrg, Phys. Rev. Lett. 79 (1997) 3795.
LETTERS A
Physics Letters A 255 (1999) 183-186
Magnetic relaxation in high-temperature superconductors Hao Jin, Lin Chen, Yu Heng Zhang Structure Research Laboratory, University of Science and Technology of China, Hefei 230026, P.R. China
Received 18 November 1998; accepted for publication 17 February 1999 Communicated by J. Flouquet
Abstract The magnetic relaxation rate was calculated for the vortex-glass or collective-creep model and the thermal activated model considering backward hopping. By comparing the theoretical results with experimental data it was found that the former models were in good agreement with the experiment at relatively low temperature regions, while the latter were at higher temperatures near T,. These results are discussed in detail. @ 1999 Elsevier Science B.V. PACS: 74.fN.Ge Keywords: relaxation rate; Flux creep; Flux dynamics; Thermal activated; Pinning; High-temperature superconductors
1. Introduction
The flux dynamics of high-temperature superconductors (HTSC) are particularly different from those of conventional superconductors. One of those important phenomena found in experiments is the non-linear relationship between M and In t, which does not fit the Kim-Anderson model [ 1,2]. Various models were proposed for explanations, including the vortex glass model by Fisher et al. [3,4], the collective-creep model by Fiegle’man et al. [ 561, the logarithmic j-dependent barrier model by Zeldov et al. [ 71, the revised Kim-Anderson model by Hagen and Griessen [ 81, and the creep model with damp by Yin et al. [ 91. All the theoretical results of these models, to some extent, can give an explanation of the experiments. In the interest of finding out more about the validity of each model, the relaxation rate s = -dM/d In t and the normalized relaxation rate = s/M were introduced so s, = -(l/M)dM/dlnt that the magnetic relaxation behavior can be described more sensitively. The experiment [ lo] showed that there was a plateau for sn in the s,-T curve from 20 K to 60 K. Theoretically, the vortex-glass or collectivecreep model gives a plateau in the s,-T relationship, which agrees with experiments. However, the logarithmic barrier model can not derive this result, and
the Kim-Anderson model gives a negative plateau result at long times and high temperatures, which does not agree with the experiments [ 111. On the other hand, experiments showed that the snT relationship exhibits a rising trend [ lo], or in the s-T relationship [ 121, s drops rapidly as T increases near T,. The reason of s,-T rising is that the dropping speed of M with T is much bigger than that of s with T. Theoretically, explanations in this temperature region have not been reported although these models have been studied for years. The relaxation rate s, rather than the normalized relaxation rate s,, is discussed in this paper while considering the following two points: first, in the vortexglass model or the collective-creep model, there is an inverse ratio relationship between sn and the parameter I(, while the s-p relationship obeys a power law and is thus more sensitive: secondly, both M and s drop as T rises, so that s, = s/M is a variable which changes slowly with T. In the present work, the relaxation rate s was derived from the vortex-glass model or the collective-creep model and compared with the experiment [ 121. Moreover, considering the relatively low activation barrier Ue in HTSC and the rather high working temperature T near T,, the contribution of backward hopping [ 13,141, which is the hopping of flux lines from low
0375-9601/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved.
184
H. Jin et al./Physics
Letters A 255 (1999) 183-186
barriers to high ones, was introduced into the thermal activated model, and the derived s was also compared with the experiment. The results suggest that the vortex-glass or the collective-creep model will dominate the flux behavior at relatively low temperatures, but as the temperature increases, especially when T is near T,, the thermal activation with both forward and backward hopping will play a more important role in the flux-relaxation process. Discussion of this are given in this paper.
2. Calculated s-T dependence 2.1. Vortex-glass model and collective-creep model
In both the vortex-glass model [3,4] and the collective-creep model [5,6] the pinning barrier satisfies the equation U(j)
2.2. Thermal activated model considering backward hopping
In the thermal activated model, while considering backward hopping, the hopping rate Y satisfies y=Zexp(
In Eq. (6), j represents the current density, B represents the average magnetic induction, d represents the size of pinning center, and vo represents the hopping rate of the initial state at which the relaxation does not happen. When t = to (Y = vg), namely at the beginning of relaxation, we have dB/dx = pajc. Also, noticing that there is no flux in the pinning center at that state, which means a forward hopping only, one obtains l=exp(
K=Zexp( where j is the electric current density, j, is the critical electric current density. The parameter ,u is less than one [ 31 for the vortex-glass model. In the collectivecreep model [ 61, p varies among values l/7, 312, 719, 918, l/2, depending on different temperatures and fields. Obviously the barrier U(j) diverges as j goes to zero. From Eq. ( 1) , using the Arrhenius relationship
t
VO
exp
U(j)
[
- -
kT
1
(2)
we have
(3) Defining the relaxation rate dM ‘= -dint
S=--dIn-=
-g)exp($)
(7)
-s)sinh($).
UO
(8)
Comparing Eq. (8) with Eq. (2)) we have U(j)=Ue-kTln(2sinh($$)).
(9)
Thus, when j ---)0, U(j)
= UO+ kTln
j&T 2juo’ ( )
(10)
It is obvious that U(j) diverges when j --t 0. Noting that j + 0 means that the Lorentz force approaches zero, so the forward and backward hopping rates become the same. These correspond to an infinite effective barrier U(j) in the Arrhenius’ equation (2). Thus the divergence of U(j) as j approaches zero has a clear physical meaning physics. Using relation Y/VIJ= to/t in Eq. (8) and solving the algebraic equation leads to
(4)
and substituting (3) into (4), one obtains dM
(6)
Combining Eqs. (6) and (7), we find
=UO[($)p11.
to y -_=-_=
-$)sinh(g).
VO
+ln[~+~/l+4~exp(-2/$)]
d(j/j=) d In t
!Jl+ $1, (-f-)I-“““.
(11)
Substituting Eq. ( 11) into Eq. (4)) we find
(5)
sfx-
kT Uo Jl
1 + 4( t/to)* exp( -2Uo/kT)
(12)
185
H. Jin et al./Physics Letters A 255 (1999) 183-186
3.0 1
U,=38.8meV
3.0
t/t,=3.8x103
lJ,=38,8meV
1/1,=3.8x103
t 2.5
0.0 0
20
40
60
80
100
T WI
T(K) Fig. 1. Calculated s-T relationship from Eq. (4) with different p. according to the vortex-glass or collective-creep model (solid lines). The stars represent the experimental data. Calculating parameters are Un = 38.8 meV, t/to = 3.8 x 103.
Fig. 2. Calculated s-T relationship from Eq. (9). according to the thermal activated model considering the backward hopping (solid line). The stars represent the experimental data. Calculating parameters are Un = 38.8 meV, t/to = 3.8 x 103.
3. Comparison of the theoretical results with the experimental data
well at high temperatures and s will go to zero as T goes to TC. However, the fit in the low temperature range is not good. The behavior of s at temperatures near T, as mentioned above is reasonable. From the physical point of view, the superconductive state goes back to a normal state when T = T, and M = 0, thus s = 0 necessarily. For type-II superconductors, more magnetic flux will penetrate the superconductors when the temperature increases, thus the proportion of the normal state part will increase, and M decreases. When T + T,,
3.1. Vortex-glass model and collective-creep model The relaxation rate s, derived from either the vortexglass model or the collective-creep model, satisfies Eq. (5), with the only difference being in the value of the parameter ,u, so s is discussed at the same time. Fig. 1 shows the calculated s-T relationship according to Eq. (5) with different ,X values. The experimental data in Fig. 1 were obtained from the work of Yeshurun, Malozemoff, Holtzberg et al. [ 121, which were measured on a YBCO single crystal at an applied field of 1 kOe. It is obvious that the temperature corresponding to the maximum s moves to the lowtemperature side as p decreases. The theories showed good agreement with the experiment at low temperatures when ,U is between l/2 and 1. But it was also noted that at high temperatures the theoretical results did not fit the experiment at all. In fact, it is obvious that s never approaches zero as T goes to T, with p ranging from l/2 to 1. When p is smaller, although the calculated s (see the ,X = l/7 curve) will reach zero at high temperatures, the fitting with experimental data is very poor over the full temperature range.
3.2. Thermal activated model considering backward hopping
The s-T relationship in Eq. ( 12) is shown in Fig. 2. As can be seen, the calculated results fit the experiment
M + 0.
Therefore it is concluded from the comparison of those models with the experiment (see Fig. 3, with parameter ,X = 0.65) that the vortex-glass model and the collective-creep model can give a good explanation of s at relatively low temperatures, while the thermal-activated model considering backward hopping can give a good explanation of s at relatively high temperatures.
4. Discussion It appears from the results mentioned above that to describe the relaxation rate s correctly in HTSC, both the vortex-glass or collective-creep model and the thermal activated model considering backward hopping should be used at different temperature regions. That is to say, there exists a cross-over from collected barriers to a single barrier as temperature rises. This conclusion is reasonable from a physical
H. Jin et al./Physics
186
3.0
Letters A 255 (1999)
.,*--*I, t
2.5
vortex
-_
/
p=odi
2.0
/’
1.5 v)
,‘I
I
1.0 -: 0.5
,I
.: _: :”
o.of 0
20
40
60
80
100
T(K) Fig. 3. Comparison between theoretical results and experimental data. The dotted line represents the vortex-glass model, the solid line represents the thermal activated model considering the hack-
ward hopping, the stars represent the experimentaldata. Calculat-
183-186
activation barrier Ua and the higher temperature (near T,) for HTSC, the contribution of flux hopping from low potentials to high ones can not be neglected. Therefore, by adding the backward flux hopping, the thermal activated model will account for the flux creep behavior at a relatively high temperature region and it is in good agreement with the experimental data. The magnetic relaxation in HTSC is a very complex question and is still not perfectly understood. We would like to mention some alternative points of view in recent work, for instance that (elastic) collective creep is valid at low temperatures or field then plastic creep wins over [ 151, or collective creep still holds but with a non-monotonous variation of the exponent ,U [ 161 until the (second-order) melting or irreversibility line is reached. Acknowledgements
ing parameters are Un = 38.8 meV, t/to = 3.8 x 103.
point of view. First, we consider the vortex-glass model to be responsible for the behavior of the magnetic relaxation at relatively low temperatures. When the temperature reaches T,, melting of the vortex lattice will occur and the system will get into the vortex liquid region. However, the pinning still exists. Hence the individual pinning in the Rim-Anderson model, rather than the vortex glass, is suitable for the description of this situation. From the experimental data, however, it can be noted that the plateau region is below the irreversibility line or the melting line. This fact is not difficult to understand, because, in general, there is not a clear boundary of transition from solid to liquid for a glass, i.e. a wide temperature range of transition should exist. Therefore, it is plausible that the cross-over may be related to the premelting softening of the vortex lattice. Secondly, according to the collective-creep model, a bundle of flux lines is pinned by several pinning centers at the low temperature region. The collective-creep model considers the case of identical barriers. In actual cases, the barriers should have some distribution. So with increasing temperature and the enhancement of thermal activation, the weaker pinning centers will be depinned. Thus when the temperature rises high enough, all other pinning centers will have no effect except for the strongest pinning site due to thermal depinning. Because of single-barrier pinning, the elastic or plastic energy then vanishes and the pinning energy will dominate the system, even that cross-over occurs in the vortex solid region [ 121. Moreover, considering the smaller
This work is financially supported by the National Center for Research & Development on Superconductivity and Foundation of Academia Sinica.
References [II Y.B. Kim, C.F. Hempstead, A.R. Stmad, Phys. Rev. Lett. 9 (1962) 306.
I21 PW. Anderson, Phys. Rev. L&t. 9 (1962) 309. [31 M.P.A. Fisher, Phys. Rev. Lett. 62 (1989) 1415. [41 M.P.A. Fisher, D.S. Fisher, D.A. Huse, Physica B 169
(1991) 85. [51 M.V. Fiegle’man, V.B. Geshkenbein, [61 [71 ISI [91 1101
[Ill [I21 [I31 1141
Cl51 [I61
A.l. Larkin, V.M. Vinokur, Phys. Rev. Lett. 63 (1989) 2303. M.V. Fiegle’man, V.B. Geshkenein, V.M. Vinokur, Phys. Rev. B 43 (1991) 6263. E. Zeldov, N.M. Amer. G. Koren et al., Phys. Rev. Lett. 62 (1989) 3093. C.W. Hagen, R. Griessen, Phys. Rev. Lett. 62 (1989) 2857. D. Yin, W. Schauer, V. Windte, H. Kupfer et al., Phys. B 94 (1994) 249. L. Civale, A.D. Marwick, M.W. McElfresh, T.K. Worthington, A.P. Malozemoff, F.H. Holtzberg, J.R. Thompson, M.A. Kirk, Phys. Rev. Lett. 65 ( 1990) 1164. Y. Yeshurun, A.P. Malozemoff, A. Shaulov, Rev. Mod. Phys. 68 (1996) 911. Y. Yeshurun, A&? Malozemoff, Phys. Rev. Lett. 60 (1988) 2202. D. Dew-Hughes, Cryogenics 28 ( 1988) 674. C.W. Hagen, R.P. Griessen, in: Studies of High Temperature Superconductors: Advances in Research and Applications, Vol. 3, Ed. A.V. Narliar (Nova Science, New York, 1990) p. 159. Y. Abulatia, A. Shaulov, Y. Wolfus, R. Prozorov, L. Burlachkov, Y. Yeshurun, Phys. Rev. Lett. 77 ( 1996) 1596. T. Klein, W. Harneit, L. Baril, C. Escribe-Filippini, D. Feinbcrg, Phys. Rev. Lett. 79 (1997) 3795.