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Exact solution of the thermally activated flux-creep model on type-II superconductors
Physics Letters A 183 (1993) 425-430 North-Holland
PHYSICS LETTERS A
Exact solution of the thermally activated flux-creep model on type-II superconductors Yong-li M a a,b,c,Xian-xi Dai b,c,dWan-heng Zhong b,c Hong-fang Li b,cand Pei-hong Hor
d
" CCAST(WorldLaboratory), P.O. Box 8730, Beijing 100080, China b Department of Physics, Fudan University, Shanghai 200433, China t T.D. Lee Physics Laboratory, Fudan University, Shanghai 200433, China d Texas Centerfor Superconductivity at UH, Houston, TX 77204-5932, USA
Received 13 July 1993; revised manuscript received20 October 1993; acceptedfor publication 22 October 1993 Communicatedby J. Flouquet
The model of thermally activated flux creep is reanalyzedtheoretically, since the approximate solution of the Anderson-Kim model is inadequate to describe the flux creep in high-Tosuperconductors. An exact solution of this model is derived. It is shown that this simple phenomenological theory best describes the magnetic relaxation data obtained in both high-To and low-T¢ superconductors.
1. Introduction
It has long been recognized that thermally activated flux creep plays an important role in the dynamic properties of type-II superconductors in a magnetic field. The thermally activated flux-creep model, first introduced by Anderson, Kim and co-workers [ 1-4 ], has been successfully applied to explain the dynamic flow of flux lines in conventional ("low-To") type-II superconductors. However, due to the extremely short coherence length and high critical temperature as compared with low-To superconductors, it is expected that high-To superconductors will have giant flux-creep effects. In addition, over a large range of intermediate temperatures the nearly temperature-independent plateau of the normalized flux relaxation rate, S = - O In J/O In t, with J the current density and t the time, is not easy to understand in this theory. To account for this plateau, many authors present various famous models with a completely different effective barrier U ( J) . For example, the inverse power law [ 5-9 ] with the effective barrier U(J)oc (Jo/J) ~ where Jo is the current density without thermal activation and/z is a power of the order of one, involving the vortex glass model [ 6,7 ] and the collective creep model [8,9], and the logarithmic law [10,11] with the barrier U ( J ) o c l n ( J / J o ) which was found by Zeldov et al. [ 10] and has been used by Vinokur et al. [ I 1 ] as a creep having a self-organized criticality in high-To superconductors. The I - Vcharacteristic then is Voc exp [ U ( J ) / k a T ] . But two problems possibly appear in these models. One is that these effective barriers apparently cannot be observed directly and their differences are large. Another is that these models lead to both the difficulty of obtaining an analytic solution of the flux-creep equation and an uncertainty of the fitting parameters [ 12 ]. Many authors [ 13-16] have pointed out that large relaxation effects for high-T¢ superconductors might arise from thermally activated flux motion, which is known to occur in conventional superconductors, though-on a much smaller scale. However, the plateau of S is hard to explain naturally. One can expect that the role of the thermal exaction is higher in high-T¢ superconductors with large values of the Ginzberg-Landau parameter, due to the higher temperature T and the smaller activation energy Uo, which changes the value regions of the ' Mailing address. 0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
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parameters such as T, Uo and its ratio kaT/Uo, etc. Since in high-To superconductors the ratio Uo/knTis considerably smaller (see fig. 1 below), reverse hops must be taken into account. Although many authors [ 1116 ] have considered the forward and reverse hops, to our knowledge, the exact solution of the flux-creep equation has not been derived so far. Dew-Hughes [ 15 ] has discussed the solution for the condition Uo << kaT, but it is impossible to satisfy this condition in high-Tc superconductors. It is both important and interesting to improve this phenomenological Anderson-Kim model and to give a consistent explanation of the experimental results. In this paper, we will reanalyze theoretically the thermally activated flux-creep model only using the simplest law of the effective barrier U(J) ocJ but considering the forward and reverse hops and find first its exact solution and explicit expression. The behavior of the flux line shows reasonable agreement with the theoretical predictions of the flux-creep model. With the exact solution the Anderson-Kim flux picture still is a suitable model for explaining type-II superconductor data (except for the quantum tunneling process of highT¢ superconductors at the lowest temperatures). This study improves our understanding of the classical flux creep.
2. The exact solution and explicit expression The flux-creep mechanism is possible due to the existence of a Lorentz-force-driven flux motion based on the Anderson-Kim model. Flux lines are held in a potential well with depth Uo (i.e., activation energy) due to both the pinning by defects and the interaction between the flux lines. According to the Lorentz force formalism, the force on a line varies linearly with the current if one assumes that the current density is constant within the sample. The well is reduced to U(J) = Uo- aJ by this driving force on the line, where a is a system parameter. Uo is the difference in the Gibbs function of the system when a flux line intersects a defect and when it is out of the defect. The well vanishes at J = J0, so from U(Jo) = 0 we can write the effective well depth as U(J) = Uo( 1-J/Jo). The I - V c u r v e then takes the form [2-4] V(J) ocexp [ - (Uo/kaT) ( 1-J/Jo) ] due to the thermal activation without the motion of the flux against the Lorentz force. From that J decays in time as OJ/Otoc - V(J) when the external field is withdrawn. This assumption leads to
J(t)=Jo{1- -~-~ln[exp(~a°T [1-J(to)/Jo])+ ~ - ] }
,
(1) --1
- -
Uo z + t - to
k a T l n exp ~
Uo
kn l
1-J(to)/Jo] +
with a hopping attempt time Z-JokBT/kUo, where k is the decay coefficient which depends on the experimental parameter such as the material parameters, the sample shape, the attempt frequency and some physical parameters. Equation ( 1 ) shows explicitly that at time to the current density is J(to). For J(to) =Jo these results are just S(T) ockBT~ Uo and J(t)oc In t in the low-T limit for conventional type-II superconductors. However, recent experiments in high-To superconductors have cast considerable doubts on the details of this theory, although the basic concept of the thermal activation remains the key to understanding the flux creep. Because the ratio Uo/kBT is considerably smaller in a large range of temperature, we must take into account the motion of the flux against the Lorentz force, whether the current density of the sample approaches zero or whether it is appreciably larger. An external current density leads to the Lorentz force on the flux lines resulting in a tilting of the barrier structure and hence to an enhancement of the hopping frequency through the energy barrier in the force direction and to a reduction in the opposite direction. The strict term sinh (JUo/JokB T) could replace the approximate term exp(JUo/JokaT) in the I - V characteristic. So we can get the exact and explicit solution with the initial condition J=J(to) at t=to, which is given by 426
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2kBT f [- t-to ( ~T)]" , [J(to)Uo'~) J(t)=Jo--~-o arctanh~expL-2 T e x p , _ _ . _ U ° tann~2j--)-~aT};"
(3)
This gives the normalized relaxation rate
S(T) =
2t e x p { - 2 [ (t-to)/Z] exp( - Uo/kBT)} tanh[J(to)Uo/2JoksT] r 1 - ( e x p { - 2 [ (t-to)/r] exp( - Uo/kBT)} tanh [J(to)Uo/2JokBT] )2
exp( - Uo/ kB T) × arctanh (exp{ - 2 [ ( t - to) / r] exp ( - Uo/kB T) ) tanh [J(to) Uo/2Jo kB T] ) "
(4)
These are our central results. For Uo >> kBT and t - to << r exp ( Uo/kn T), (3) and (4) turn into ( 1 ) and (2) respectively. For t>> to, r, and J(to) =Jo, ( I ) and (2) turn further into
J(t) =Jo[1 - (knT/Uo) In(t/q) ]
(5)
and
knr/Uo S(T) = 1 - (kBT/Uo) I n ( t / r ) '
(6)
which are just the usual Anderson-Kim formulas [ 1-4].
3. Results and discussions
Define the reduced time a = k(t-to)IJo and the reduced activation energy fl= UolkBT. We find that J ( t ) / Jo in (3) and S(T)( 1-to~t) in (4) can be expressed by only these two parameters. Although Jo and k are hard to measure and t varies by several orders of magnitude in experiments, S ( T ) and J(t) are insensitive to ao which is defined as Oto=Xto/Jo. A typical value of Oto is 0.015 after fitting some experimental data at the time window 10-4-101 s [12]. Before investigating the detailed dependence of S ( T ) on T, we first consider the curve of S ( T ) as a function of]/-1 with a certain value of ot for both type-II superconductors. From (4) with a = 0.015 [ 17 ], the typical result is shown in fig. 1 for high-To superconductors. The inset is for low-To
0.03
0.03
0.02
0.01
I
0.00 0.0
0.00
' 0.1
0.2
0.3
KaT/Uo Fig. 1. Normalized time-logarithmic current density decay rate S -= - # in J ( t ) / a In t versus the ratio kBT/Uo(T) for high-To superconductors. The inset shows that for low-To superconductors.
0
' I 0
~ 20
4
' 30
T(K)
' 40
50
60
Fig. 2. Normalized time-logarithmic current density decay rate S-= - O In J ( t ) / O In t versus temperature T for the high-To superconductor YiBa2Cu307 with To= 90 K. The inset shows that for low-To superconductors N b - Z r with To= 10 K.
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superconductors at sufficiently long measurement t, c~>> 1. Figure 1 shows a plateau over a large range of kBT/ Uo with a downward trend at smaller values of kBT/Uo. Using this curve, we are able to determine Uo(T) experimentally within the framework of the thermally activated flux-creep model. We now consider the curve of S ( T ) versus T. From a more fundamental physical model [ 18 ] and a phenomenological theory of layered high-T¢ superconductors [ 19] we know that the strictly analytical form of Uo(T) is inversely proportional to the penetration depth 2 (T), and the approximately analytical form of 2 (T) is inversely proportional to the square root of 1 - T ~ T¢. One therefore can write Uo(T) = Uo(0) ( 1 - T/Tc)1/2.
(7)
Choosing [4,16,20,21 ] U o ( 0 ) = 2 0 meV, o~=0.015 and To=90 K for the Y i B a 2 C u 3 0 7 superconductor, fig. 2 gives the curve of S ( T ) calculated from (4) and (7). It can be seen that the curve is usually divided into three regions: a flat maximum over a large range of intermediate temperatures, a nonlinear curvature at a lower temperature range, and a rapid vanishing at the lowest temperatures. The first two regions are in reasonable agreement with the experimental data given in refs. [20,21 ] by reviewers while the unexpected last region is clearly an inconsistency which has been solved in ref. [ 22 ] by considering the quantum tunneling effects in the quantum regime. Notice that 2 (T)oc [1 - (T~ T~)4]-1/2 for low-Tc superconductors, one can write
Uo(T) = Uo(O) [1 - (T/Tc) 4] t/2
(8)
In the experiments of Kim et al. [ 1,3 ] for the low-To superconductor N b - Z r (No. 12D) with To-- 10.0 K, the value of S at T = 4 . 2 K is about 1/600. From (4) and (8), one obtains Uo(0)~220.6 meV. Since the temperature dependence of S ( T ) remains linear down to T~ To= 0.1 [1,3], S ( T ) at T = 1.0 K is determined from (6) and ( 8 ) to be 0.0004. So choosing Uo (0) -- 220 meV, ot >> 1 and T¢-- 10 K for the N b - Z r superconductor, the inset of fig. 2 gives the curve of S ( T ) calculated from (4) and (8). It is just that S(T)ocTdue to the low T and large Uo. Figure 2 suggests that the exact solution (4) is universally applicable to a great variety of relaxation rates within the framework of the thermally activated flux-creep model, while the approximate solution (6) is only appropriate for low T and large U0. This is obvious due to Uo~ nkB T (where n is a number of the order often) for high-To superconductors in which the reverse creep cannot be ignored and due to Uo >> kaT for low-To superconductors in which the reverse creep is indeed negligible. We finally consider the curves of J ( T ) versus Int. The curves of the reduced current density J(t)/J(to) as a function of the reduced logarithmic time ln(t/to) calculated from (3) with J(to)=Jo are shown in fig. 3. The solid line is for high-T¢ superconductors at T = 2 0 K with Uo(20)-- 17 meV, while the dashed line is for low- T¢ superconductors at T = 5 K with Uo (0) = 200 meV. Figure 3 exhibits a nonlinear logarithmic time decay of the current density over both a long and a short time for high-T~ superconductors. These behaviors have been observed experimentally [ 23,24 ]. But for low-T¢ superconductors the decay of the current density is slowly 1.0
o
~o.s
0.0 4
428
In(Uto)
8
12
Fig. 3. Reduced current density J/J(to) versus reduced logarithmic time In (t/to) for high- Tc superconductors at T = 20 K with Uo ( 2 0 ) = 17 meV (solid line). The dotted line presents the linear one. The dashed line shows that for low-T¢ superconductors at T = 5 K with Uo (0) = 200 meV.
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linear in In t. This is reasonable due to Uo ~ nkBT for high-To superconductors a n d due to Uo >> kBT for lowT~ superconductors.
4. Concluding remarks In s u m m a r y , we have reanalyzed theoretically the m o d e l o f thermally activated flux creep a n d found its exact a n d explicit solutions. The theoretical curves are reasonably consistent with the experimental observations in b o t h superconductors for all t e m p e r a t u r e s except for the lowest temperatures o f the high-To superconductors. Although b o t h superconductors have different properties o f the current-density decay, they are both based on the same physical foundation. C o m p a r i n g the giant flux-creep effect on high-To superconductors with the small one on low-To superconductors, we conclude that the reverse hops b e c o m e i m p o r t a n t due to a high T and a small Uo. It leads to a slow e n h a n c e m e n t o f the hops in the Lorentz force direction and to a fast reduction in the opposite direction and hence the decay rates S saturate, so we emphasize the i m p o r t a n c e o f the reverse hops. We also conclude that it is b o t h essential a n d useful to solve the flux-creep equation exactly. However, the concept o f thermally activated flux creep a n d the simple phenomenological theory resulting from this concept are very effective in unifying various p h e n o m e n a associated with the m i x e d state o f type-II superconductors, although the region o f lowest temperatures needs some other description. At least for the understanding o f the magnetization relaxation, the m o r e elaborate pinning theories are not needed, as long as one knows the t e m p e r a t u r e d e p e n d e n c e Uo ( T ) . This m o d e l in which one assumes that the effective barrier remains finite as J decays is m o r e acceptable than other models in which U ( J - - . 0 ) ~ ~ in a different way. This exactly solvable m o d e l for simplicity is explicitly expressed by only ot and ft. The details o f the fitting parameters and hence the d e t e r m i n a t i o n o f all p a r a m e t e r s by c o m b i n e d t h e o r e t i c a l - e x p e r i m e n t a l curves will be given in further publications.
Acknowledgement This work is s u p p o r t e d in part b y the Chinese F o u n d a t i o n o f Doctoral Education and by the Shanghai F o u n d a t i o n o f N a t u r a l Science. One o f the authors (X. D a i ) would like to express his sincere thanks for the support o f N S F G r a n t No. D M R 91-22043, D A R P A G r a n t No. M D A 972-88-G-002, the State o f Texas, NSAS G r a n t No. NAQW-977, the T.L.L. Temple F o u n d a t i o n a n d the Texas Center for Superconductivity at the University o f Houston.
References [ 1] Y.B. Kim, C.F. Hempstead and A.R. Strand, Phys. Rev. Lett. 9 (1962) 306. [ 2 ] P.W. Anderson, Phys. Rev. Lett. 9 (1962) 309. [ 3 ] Y.B. Kim, C.F. Hempstead and A.R. Strand, Phys. Rev. 129 (1963) 528; 131 ( 1963 ) 2486. [4] P.W. Anderson and Y.B. Kim, Rev. Mod. Phys. 36 (1964) 39. [ 5 ] E. Zeldov, N.M. Amer, G. Koren, A. Gupta, M.W. McElfresh and R.J. Gambino, Appl. Phys. Lett. 56 (1990) 580. [6] M.P.A. Fisher, Phys. Rev. Lett. 62 (1989) 1415. [ 7 ] D.S. Fisher, M.P.A. Fisher and D.A. Huse, Phys. Rev. B 43 ( 1991 ) 130, and references therein. [8 ] M.V. Feigol'man, V.B. Geshkenbein, A.L. Larkin and V.M. Vinokur, Phys. Rev. Lett. 63 ( 1989 ) 2303; M.V. Feigol'man, V.B. Geshkenbein and V.M. Vinokur, Phys. Rev. B 43 ( 1991 ) 6263. [ 9 ] ICH. Fischer and T. Nattermann, Phys. Rev. B 43 ( 1991 ) 10372, and references therein. [ 10 ] E. Zeldov, N.M. Amer, G. Koren, A. Gupta, R.J. Gambino and M.W. McElfresh, Phys. Rev. Left. 62 (1989) 3093. [ 11 ] V.M. Vinokur, M.V. Feigol'man and V.G. Geshkenbein, Phys. Rev. Lett. 67 ( 1991 ) 915; 68 (1992) 2707; H.G. Schnack and R. Griessen, Phys. Rev. Lett. 68 (1992) 2706. 429
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[ 12 ] X.X. Dai, P.H. Hor, L. Gao and C.W. Chu, Phys. Lett. A 169 (1992) 161. [ 13 ] C.W. Hagen, R. Griessen and E. Salomons, Physica C 157 (1989) 1991. [ 14] R. Griessen, Phys. Rev. Lett. 64 (1990) 1674. [ 15 ] D. Dew-Hughes, Cryogenics 28 (1988) 674. [ 16] D. Shi and S. Jr. Salem-Sugu, Phys. Rev. B 44 ( 1991 ) 7647. [ 17 ] L. Fruchter, A.P. Malozemoff, I.A. Campbell, J. Sanchez, M. Konczkowski, R. Griessen and F. Holtzberg, Phys. Rev. B 49 ( 1991 ) 8709. [ 18 ] Y.L. Ma, H.F. Li and X.X. Dai, Phys. Rev. B 47 ( 1993 ) 12304. [ 19 ] Y.L. Ma and X.X. Dai, Int. J. Mod. Phys. B 6 ( 1992 ) 2499. [20 ] A.P. Malozemoff and M.P.A. Fisher, Phys. Rev. B 42 (1990) 6784. [ 21 ] A.P. Malozemoff, Physica C 185-187 ( 1991 ) 264. [22] Y.L. Ma, X.X. Dai, W.H. Zhong, H.F. Li and P.H. Hor, Phys. Lett. A 174 (1993) 139. [23] D. Shi and M. Xu, Phys. Rev. B 44 ( 1991 ) 4548. [24] M.D. Len, J.Z. Lou and R.N. Shelton, Phys. Rev. B 44 ( 1991 ) 2751.
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PHYSICS LETTERS A
Exact solution of the thermally activated flux-creep model on type-II superconductors Yong-li M a a,b,c,Xian-xi Dai b,c,dWan-heng Zhong b,c Hong-fang Li b,cand Pei-hong Hor
d
" CCAST(WorldLaboratory), P.O. Box 8730, Beijing 100080, China b Department of Physics, Fudan University, Shanghai 200433, China t T.D. Lee Physics Laboratory, Fudan University, Shanghai 200433, China d Texas Centerfor Superconductivity at UH, Houston, TX 77204-5932, USA
Received 13 July 1993; revised manuscript received20 October 1993; acceptedfor publication 22 October 1993 Communicatedby J. Flouquet
The model of thermally activated flux creep is reanalyzedtheoretically, since the approximate solution of the Anderson-Kim model is inadequate to describe the flux creep in high-Tosuperconductors. An exact solution of this model is derived. It is shown that this simple phenomenological theory best describes the magnetic relaxation data obtained in both high-To and low-T¢ superconductors.
1. Introduction
It has long been recognized that thermally activated flux creep plays an important role in the dynamic properties of type-II superconductors in a magnetic field. The thermally activated flux-creep model, first introduced by Anderson, Kim and co-workers [ 1-4 ], has been successfully applied to explain the dynamic flow of flux lines in conventional ("low-To") type-II superconductors. However, due to the extremely short coherence length and high critical temperature as compared with low-To superconductors, it is expected that high-To superconductors will have giant flux-creep effects. In addition, over a large range of intermediate temperatures the nearly temperature-independent plateau of the normalized flux relaxation rate, S = - O In J/O In t, with J the current density and t the time, is not easy to understand in this theory. To account for this plateau, many authors present various famous models with a completely different effective barrier U ( J) . For example, the inverse power law [ 5-9 ] with the effective barrier U(J)oc (Jo/J) ~ where Jo is the current density without thermal activation and/z is a power of the order of one, involving the vortex glass model [ 6,7 ] and the collective creep model [8,9], and the logarithmic law [10,11] with the barrier U ( J ) o c l n ( J / J o ) which was found by Zeldov et al. [ 10] and has been used by Vinokur et al. [ I 1 ] as a creep having a self-organized criticality in high-To superconductors. The I - Vcharacteristic then is Voc exp [ U ( J ) / k a T ] . But two problems possibly appear in these models. One is that these effective barriers apparently cannot be observed directly and their differences are large. Another is that these models lead to both the difficulty of obtaining an analytic solution of the flux-creep equation and an uncertainty of the fitting parameters [ 12 ]. Many authors [ 13-16] have pointed out that large relaxation effects for high-T¢ superconductors might arise from thermally activated flux motion, which is known to occur in conventional superconductors, though-on a much smaller scale. However, the plateau of S is hard to explain naturally. One can expect that the role of the thermal exaction is higher in high-T¢ superconductors with large values of the Ginzberg-Landau parameter, due to the higher temperature T and the smaller activation energy Uo, which changes the value regions of the ' Mailing address. 0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
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parameters such as T, Uo and its ratio kaT/Uo, etc. Since in high-To superconductors the ratio Uo/knTis considerably smaller (see fig. 1 below), reverse hops must be taken into account. Although many authors [ 1116 ] have considered the forward and reverse hops, to our knowledge, the exact solution of the flux-creep equation has not been derived so far. Dew-Hughes [ 15 ] has discussed the solution for the condition Uo << kaT, but it is impossible to satisfy this condition in high-Tc superconductors. It is both important and interesting to improve this phenomenological Anderson-Kim model and to give a consistent explanation of the experimental results. In this paper, we will reanalyze theoretically the thermally activated flux-creep model only using the simplest law of the effective barrier U(J) ocJ but considering the forward and reverse hops and find first its exact solution and explicit expression. The behavior of the flux line shows reasonable agreement with the theoretical predictions of the flux-creep model. With the exact solution the Anderson-Kim flux picture still is a suitable model for explaining type-II superconductor data (except for the quantum tunneling process of highT¢ superconductors at the lowest temperatures). This study improves our understanding of the classical flux creep.
2. The exact solution and explicit expression The flux-creep mechanism is possible due to the existence of a Lorentz-force-driven flux motion based on the Anderson-Kim model. Flux lines are held in a potential well with depth Uo (i.e., activation energy) due to both the pinning by defects and the interaction between the flux lines. According to the Lorentz force formalism, the force on a line varies linearly with the current if one assumes that the current density is constant within the sample. The well is reduced to U(J) = Uo- aJ by this driving force on the line, where a is a system parameter. Uo is the difference in the Gibbs function of the system when a flux line intersects a defect and when it is out of the defect. The well vanishes at J = J0, so from U(Jo) = 0 we can write the effective well depth as U(J) = Uo( 1-J/Jo). The I - V c u r v e then takes the form [2-4] V(J) ocexp [ - (Uo/kaT) ( 1-J/Jo) ] due to the thermal activation without the motion of the flux against the Lorentz force. From that J decays in time as OJ/Otoc - V(J) when the external field is withdrawn. This assumption leads to
J(t)=Jo{1- -~-~ln[exp(~a°T [1-J(to)/Jo])+ ~ - ] }
,
(1) --1
- -
Uo z + t - to
k a T l n exp ~
Uo
kn l
1-J(to)/Jo] +
with a hopping attempt time Z-JokBT/kUo, where k is the decay coefficient which depends on the experimental parameter such as the material parameters, the sample shape, the attempt frequency and some physical parameters. Equation ( 1 ) shows explicitly that at time to the current density is J(to). For J(to) =Jo these results are just S(T) ockBT~ Uo and J(t)oc In t in the low-T limit for conventional type-II superconductors. However, recent experiments in high-To superconductors have cast considerable doubts on the details of this theory, although the basic concept of the thermal activation remains the key to understanding the flux creep. Because the ratio Uo/kBT is considerably smaller in a large range of temperature, we must take into account the motion of the flux against the Lorentz force, whether the current density of the sample approaches zero or whether it is appreciably larger. An external current density leads to the Lorentz force on the flux lines resulting in a tilting of the barrier structure and hence to an enhancement of the hopping frequency through the energy barrier in the force direction and to a reduction in the opposite direction. The strict term sinh (JUo/JokB T) could replace the approximate term exp(JUo/JokaT) in the I - V characteristic. So we can get the exact and explicit solution with the initial condition J=J(to) at t=to, which is given by 426
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2kBT f [- t-to ( ~T)]" , [J(to)Uo'~) J(t)=Jo--~-o arctanh~expL-2 T e x p , _ _ . _ U ° tann~2j--)-~aT};"
(3)
This gives the normalized relaxation rate
S(T) =
2t e x p { - 2 [ (t-to)/Z] exp( - Uo/kBT)} tanh[J(to)Uo/2JoksT] r 1 - ( e x p { - 2 [ (t-to)/r] exp( - Uo/kBT)} tanh [J(to)Uo/2JokBT] )2
exp( - Uo/ kB T) × arctanh (exp{ - 2 [ ( t - to) / r] exp ( - Uo/kB T) ) tanh [J(to) Uo/2Jo kB T] ) "
(4)
These are our central results. For Uo >> kBT and t - to << r exp ( Uo/kn T), (3) and (4) turn into ( 1 ) and (2) respectively. For t>> to, r, and J(to) =Jo, ( I ) and (2) turn further into
J(t) =Jo[1 - (knT/Uo) In(t/q) ]
(5)
and
knr/Uo S(T) = 1 - (kBT/Uo) I n ( t / r ) '
(6)
which are just the usual Anderson-Kim formulas [ 1-4].
3. Results and discussions
Define the reduced time a = k(t-to)IJo and the reduced activation energy fl= UolkBT. We find that J ( t ) / Jo in (3) and S(T)( 1-to~t) in (4) can be expressed by only these two parameters. Although Jo and k are hard to measure and t varies by several orders of magnitude in experiments, S ( T ) and J(t) are insensitive to ao which is defined as Oto=Xto/Jo. A typical value of Oto is 0.015 after fitting some experimental data at the time window 10-4-101 s [12]. Before investigating the detailed dependence of S ( T ) on T, we first consider the curve of S ( T ) as a function of]/-1 with a certain value of ot for both type-II superconductors. From (4) with a = 0.015 [ 17 ], the typical result is shown in fig. 1 for high-To superconductors. The inset is for low-To
0.03
0.03
0.02
0.01
I
0.00 0.0
0.00
' 0.1
0.2
0.3
KaT/Uo Fig. 1. Normalized time-logarithmic current density decay rate S -= - # in J ( t ) / a In t versus the ratio kBT/Uo(T) for high-To superconductors. The inset shows that for low-To superconductors.
0
' I 0
~ 20
4
' 30
T(K)
' 40
50
60
Fig. 2. Normalized time-logarithmic current density decay rate S-= - O In J ( t ) / O In t versus temperature T for the high-To superconductor YiBa2Cu307 with To= 90 K. The inset shows that for low-To superconductors N b - Z r with To= 10 K.
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superconductors at sufficiently long measurement t, c~>> 1. Figure 1 shows a plateau over a large range of kBT/ Uo with a downward trend at smaller values of kBT/Uo. Using this curve, we are able to determine Uo(T) experimentally within the framework of the thermally activated flux-creep model. We now consider the curve of S ( T ) versus T. From a more fundamental physical model [ 18 ] and a phenomenological theory of layered high-T¢ superconductors [ 19] we know that the strictly analytical form of Uo(T) is inversely proportional to the penetration depth 2 (T), and the approximately analytical form of 2 (T) is inversely proportional to the square root of 1 - T ~ T¢. One therefore can write Uo(T) = Uo(0) ( 1 - T/Tc)1/2.
(7)
Choosing [4,16,20,21 ] U o ( 0 ) = 2 0 meV, o~=0.015 and To=90 K for the Y i B a 2 C u 3 0 7 superconductor, fig. 2 gives the curve of S ( T ) calculated from (4) and (7). It can be seen that the curve is usually divided into three regions: a flat maximum over a large range of intermediate temperatures, a nonlinear curvature at a lower temperature range, and a rapid vanishing at the lowest temperatures. The first two regions are in reasonable agreement with the experimental data given in refs. [20,21 ] by reviewers while the unexpected last region is clearly an inconsistency which has been solved in ref. [ 22 ] by considering the quantum tunneling effects in the quantum regime. Notice that 2 (T)oc [1 - (T~ T~)4]-1/2 for low-Tc superconductors, one can write
Uo(T) = Uo(O) [1 - (T/Tc) 4] t/2
(8)
In the experiments of Kim et al. [ 1,3 ] for the low-To superconductor N b - Z r (No. 12D) with To-- 10.0 K, the value of S at T = 4 . 2 K is about 1/600. From (4) and (8), one obtains Uo(0)~220.6 meV. Since the temperature dependence of S ( T ) remains linear down to T~ To= 0.1 [1,3], S ( T ) at T = 1.0 K is determined from (6) and ( 8 ) to be 0.0004. So choosing Uo (0) -- 220 meV, ot >> 1 and T¢-- 10 K for the N b - Z r superconductor, the inset of fig. 2 gives the curve of S ( T ) calculated from (4) and (8). It is just that S(T)ocTdue to the low T and large Uo. Figure 2 suggests that the exact solution (4) is universally applicable to a great variety of relaxation rates within the framework of the thermally activated flux-creep model, while the approximate solution (6) is only appropriate for low T and large U0. This is obvious due to Uo~ nkB T (where n is a number of the order often) for high-To superconductors in which the reverse creep cannot be ignored and due to Uo >> kaT for low-To superconductors in which the reverse creep is indeed negligible. We finally consider the curves of J ( T ) versus Int. The curves of the reduced current density J(t)/J(to) as a function of the reduced logarithmic time ln(t/to) calculated from (3) with J(to)=Jo are shown in fig. 3. The solid line is for high-T¢ superconductors at T = 2 0 K with Uo(20)-- 17 meV, while the dashed line is for low- T¢ superconductors at T = 5 K with Uo (0) = 200 meV. Figure 3 exhibits a nonlinear logarithmic time decay of the current density over both a long and a short time for high-T~ superconductors. These behaviors have been observed experimentally [ 23,24 ]. But for low-T¢ superconductors the decay of the current density is slowly 1.0
o
~o.s
0.0 4
428
In(Uto)
8
12
Fig. 3. Reduced current density J/J(to) versus reduced logarithmic time In (t/to) for high- Tc superconductors at T = 20 K with Uo ( 2 0 ) = 17 meV (solid line). The dotted line presents the linear one. The dashed line shows that for low-T¢ superconductors at T = 5 K with Uo (0) = 200 meV.
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linear in In t. This is reasonable due to Uo ~ nkBT for high-To superconductors a n d due to Uo >> kBT for lowT~ superconductors.
4. Concluding remarks In s u m m a r y , we have reanalyzed theoretically the m o d e l o f thermally activated flux creep a n d found its exact a n d explicit solutions. The theoretical curves are reasonably consistent with the experimental observations in b o t h superconductors for all t e m p e r a t u r e s except for the lowest temperatures o f the high-To superconductors. Although b o t h superconductors have different properties o f the current-density decay, they are both based on the same physical foundation. C o m p a r i n g the giant flux-creep effect on high-To superconductors with the small one on low-To superconductors, we conclude that the reverse hops b e c o m e i m p o r t a n t due to a high T and a small Uo. It leads to a slow e n h a n c e m e n t o f the hops in the Lorentz force direction and to a fast reduction in the opposite direction and hence the decay rates S saturate, so we emphasize the i m p o r t a n c e o f the reverse hops. We also conclude that it is b o t h essential a n d useful to solve the flux-creep equation exactly. However, the concept o f thermally activated flux creep a n d the simple phenomenological theory resulting from this concept are very effective in unifying various p h e n o m e n a associated with the m i x e d state o f type-II superconductors, although the region o f lowest temperatures needs some other description. At least for the understanding o f the magnetization relaxation, the m o r e elaborate pinning theories are not needed, as long as one knows the t e m p e r a t u r e d e p e n d e n c e Uo ( T ) . This m o d e l in which one assumes that the effective barrier remains finite as J decays is m o r e acceptable than other models in which U ( J - - . 0 ) ~ ~ in a different way. This exactly solvable m o d e l for simplicity is explicitly expressed by only ot and ft. The details o f the fitting parameters and hence the d e t e r m i n a t i o n o f all p a r a m e t e r s by c o m b i n e d t h e o r e t i c a l - e x p e r i m e n t a l curves will be given in further publications.
Acknowledgement This work is s u p p o r t e d in part b y the Chinese F o u n d a t i o n o f Doctoral Education and by the Shanghai F o u n d a t i o n o f N a t u r a l Science. One o f the authors (X. D a i ) would like to express his sincere thanks for the support o f N S F G r a n t No. D M R 91-22043, D A R P A G r a n t No. M D A 972-88-G-002, the State o f Texas, NSAS G r a n t No. NAQW-977, the T.L.L. Temple F o u n d a t i o n a n d the Texas Center for Superconductivity at the University o f Houston.
References [ 1] Y.B. Kim, C.F. Hempstead and A.R. Strand, Phys. Rev. Lett. 9 (1962) 306. [ 2 ] P.W. Anderson, Phys. Rev. Lett. 9 (1962) 309. [ 3 ] Y.B. Kim, C.F. Hempstead and A.R. Strand, Phys. Rev. 129 (1963) 528; 131 ( 1963 ) 2486. [4] P.W. Anderson and Y.B. Kim, Rev. Mod. Phys. 36 (1964) 39. [ 5 ] E. Zeldov, N.M. Amer, G. Koren, A. Gupta, M.W. McElfresh and R.J. Gambino, Appl. Phys. Lett. 56 (1990) 580. [6] M.P.A. Fisher, Phys. Rev. Lett. 62 (1989) 1415. [ 7 ] D.S. Fisher, M.P.A. Fisher and D.A. Huse, Phys. Rev. B 43 ( 1991 ) 130, and references therein. [8 ] M.V. Feigol'man, V.B. Geshkenbein, A.L. Larkin and V.M. Vinokur, Phys. Rev. Lett. 63 ( 1989 ) 2303; M.V. Feigol'man, V.B. Geshkenbein and V.M. Vinokur, Phys. Rev. B 43 ( 1991 ) 6263. [ 9 ] ICH. Fischer and T. Nattermann, Phys. Rev. B 43 ( 1991 ) 10372, and references therein. [ 10 ] E. Zeldov, N.M. Amer, G. Koren, A. Gupta, R.J. Gambino and M.W. McElfresh, Phys. Rev. Left. 62 (1989) 3093. [ 11 ] V.M. Vinokur, M.V. Feigol'man and V.G. Geshkenbein, Phys. Rev. Lett. 67 ( 1991 ) 915; 68 (1992) 2707; H.G. Schnack and R. Griessen, Phys. Rev. Lett. 68 (1992) 2706. 429
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[ 12 ] X.X. Dai, P.H. Hor, L. Gao and C.W. Chu, Phys. Lett. A 169 (1992) 161. [ 13 ] C.W. Hagen, R. Griessen and E. Salomons, Physica C 157 (1989) 1991. [ 14] R. Griessen, Phys. Rev. Lett. 64 (1990) 1674. [ 15 ] D. Dew-Hughes, Cryogenics 28 (1988) 674. [ 16] D. Shi and S. Jr. Salem-Sugu, Phys. Rev. B 44 ( 1991 ) 7647. [ 17 ] L. Fruchter, A.P. Malozemoff, I.A. Campbell, J. Sanchez, M. Konczkowski, R. Griessen and F. Holtzberg, Phys. Rev. B 49 ( 1991 ) 8709. [ 18 ] Y.L. Ma, H.F. Li and X.X. Dai, Phys. Rev. B 47 ( 1993 ) 12304. [ 19 ] Y.L. Ma and X.X. Dai, Int. J. Mod. Phys. B 6 ( 1992 ) 2499. [20 ] A.P. Malozemoff and M.P.A. Fisher, Phys. Rev. B 42 (1990) 6784. [ 21 ] A.P. Malozemoff, Physica C 185-187 ( 1991 ) 264. [22] Y.L. Ma, X.X. Dai, W.H. Zhong, H.F. Li and P.H. Hor, Phys. Lett. A 174 (1993) 139. [23] D. Shi and M. Xu, Phys. Rev. B 44 ( 1991 ) 4548. [24] M.D. Len, J.Z. Lou and R.N. Shelton, Phys. Rev. B 44 ( 1991 ) 2751.
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