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A theory of thermally activated flux creep in nonideal type II superconductors
Physica C 159 (1989) 743-759 North-Holland, Amsterdam
A T H E O R Y O F T H E R M A L L Y A C T I V A T E D F L U X C R E E P I N N O N I D E A L T Y P E II SUPERCONDUCTORS
K. YAMAFUJI, T. FUJIYOSHI, K. T O K O and T. M A T S U S H I T A Department of Electronics, Kyushu University 36 Fukuoka 812, Japan Received 26 may 1989
The relaxation of the magnetization due to thermally activated flux creep in a bulk sample of a nonideal type II superconductor is discussed theoretically. Firstly, an expression for the average volume of a flux bundle characterizing its coherent thermal hopping motion is derived by starting with an equation for the local force balance on pinned fluxoids. Secondly, a thermal hopping rate of flux bundles is derived from the Fokker-Planck equation of pinned flux bundles. These results are used to derive expressions for both the time dependence of the magnetization and the induced electric field due to flux creep. It is shown that the present theory can quantitatively explain the existing observed data of the flux-creep rate, not only in conventional superconductors such as Nb-Ta and Pb-TI but also in strongly-pinning bulk samples of the high-Tooxide superconductors such as Y-Ba-Cu-O.
1. Introduction It has been p o i n t e d out that t h e r m a l l y - a c t i v a t e d flux-creep rates in high-To oxide superconductors are rem a r k a b l y larger than those in conventional superconductors, especially at higher t e m p e r a t u r e s [ 1-6 ]. This has been a t t r i b u t e d to the following sources: i) thermal fluctuation is stronger at higher t e m p e r a t u r e s a n d ii) pinning forces in existing high-To oxide superconductors are much weaker than those in conventional superconductors p r e s u m a b l y because o f the very short coherence lengths o f the high-To superconductors [ 1,7 ]. In conventional nonideal type II superconductors, the effects o f thermally-activated flux creep on their practical applications have been considered as not very serious a r o u n d 4.2 K [ 8 ]. While a recently observed fluxcreep rate at 77 K in a fairly strongly pinning bulk sample o f Y - B a - C u - O [9 ] is very small c o m p a r e d to the flux-creep rate in the so-called single crystals o f the high-T~ oxide superconductors, it is still possible that the effects o f t h e r m a l l y - a c t i v a t e d flux creep should be taken into account in detail when designing superconducting devices using high-Tc oxide superconductors. Whereas several theories have been presented on thermally-act i v a t e d flux creep in nonideal type II superconductors [ 8 , 1 0 - I 3], none o f these theories can explain quantitatively the existing o b s e r v e d d a t a even in conventional superconductors. It is, therefore, desirable to present a quantitative theory o f t h e r m a l l y - a c t i v a t e d flux creep in bulk samples o f conventional superconductors as well as o f high-T~ oxide superconductors, at least for strongly-pinning bulk samples which m a y be usable for practical applications. In this paper, we start with so-called Labusch equation [ 14 ] which describes the local force-balance on pinned fluxoids and which has been used to derive expressions for the semi-macroscopic pinning force density, Fp = B X J~ [ 14-16 ]. After m a k i n g a slight m o d i f i c a t i o n o f this equation so as to describe the local Lorentz-force density exactly, we derive expressions for correlation lengths o f p i n n e d fluxoids u n d e r thermal fluctuation forces. Since this sort o f correlated v o l u m e o f p i n n e d fluxoids gives an average v o l u m e o f each flux bundle characterizing its coherent thermal h o p p i n g motion, we derive a local force-balance equation o f flux bundles by integrating the starting equation over the average v o l u m e o f a flux bundle. This type o f the F o k k e r - P l a n c k equation [ 17 ] d e r i v e d from this local force-balance equation is known as the Smoluchowski equation [ 18 ], a n d has been used 0 9 2 1 - 4 5 3 4 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
744
K. Yamafuji et aL / Thermally activatedflux creep
to derive the thermal hopping rate of particles across potential peaks. The resulting expression for the thermal hopping rate of flux bundles across the summed-up pinning potential is used to derive an equation describing the semi-macroscopic flux density, B(r, t). The combination of this equation and the critical-state model [8,19 ] leads to a final expression for the thermally-activated flux-creep rate in a bulk sample of nonideal type II superconductors. Since the present expression for the fux-creep rate is a given function of observable physical quantities only, the present expression can be compared quantitatively with observed data. It is shown that the present theory can explain the existing observed data quantitatively, as far as fairly strongly-pinning bulk samples including high-Tc oxide superconductors are concerned.
2. Average volume of flux bundles Let us approximate the periodic fluxoid lattice in the absence of pins as a continuum with an elastic modulus tensor 1~ [20,21 ]. If pinning centers are introduced at the positions r=rn ( n = 1, 2, ... ,Np) inside the bulk sample, a volume element of the fluxoid continuum at the position r suffers a displacement uo(r). Since the change of ~ due to the introduction of pinning centers is usually negligible, the equation representing the local force-density balance can be expressed as
[)2'Uo- ~ VU,(r+uo-r,,)=O, pt
(2.1)
where U p ( r - r n ) is the elementary pinning potential and D2 is defined as [14]
02 02 a2 CII ~ -J-C66 ~ q- C44 OZ-'~" D2
02 (C11 -- C66)axay
02 02 02 C66~x2 ~-Cll--0y2 Jr-C44 ~z2
02 (Cii--C66)
(2.2)
In eqs. (2.1) and (2.2), the summation with respect to n is taken over a unit volume and the z-axis is chosen along the direction of the fluxoids. Next, let us apply a local transport current and let J represent the transport current density averaged over a semi-macroscopic volume in which the flux density B can be regarded as nearly constant with respect to r. Then the displacement uj(r, t) of the volume element at the position r at time t caused by the presence of J obeys the local force-density balance given by ((OUg/Ot) =D2" (Ug +Uo) - ~ V U v ( r + u o + u g - r n ) ,
(2.3)
n
where ( is the friction coefficient of fiuxoids per unit volume. The semi-macroscopic Lorentz-force density, F L = J × B , can be defined as the mean distorting force density obtained by averaging if2 "Us over the above-mentioned semi-macroscopic volume. Then we can formally express F L by FL = ( ~)2 "Uj ) = D2 "Ujo ,
(2.4)
where ( ) denotes the averaging over the above-mentioned semi-macroscopic volume, and Ujo(r, t) is a systematic displacement of fluxoids without shear induced by the application of the transport current. Then we can define the relative displacement u(r, t) as a deviation of uj from Ujo by
K. Yamafuji et al. / Thermally activatedflux creep
u=us-ujo
;
(u)=0.
745 (2.5)
When J is smaller than the so-called critical current density Jc, no flux flow occurs so that the magnitude of uj does not vary appreciably with time. If we confine the present discussion to the case of usual pinning strength in which no large plastic deformation of the fluxoid lattice is induced by an introduction of pinning centers, Up in eq. (2.3) can be expanded with respect to us. Then the average of eq. (2.3) over the above-mentioned semi-macroscopic volume around the position r leads to
~( OUjo/Ot) =FL -- ( ( E V2UpI -Jr~l ~ V4Uplu2o )UJO.
(2.6)
Insertion of eq. ( 2 . 6 ) i n t o eq. ( 2 . 3 ) y i e l d s '
((Ou/Ot) = 0 2 . u - a ( u ; Ujo)u+R o (r,
t) ;
(2.7a)
Og(/,/~/,/j0)~ ( E V2Up/ (..~1 E v 4 g p / (u20 "9ffu2) '
(2.8)
where Rp(r, t) is a remaining "fluctuation" pinning force defined by ( R p ) =0. Since the effect of Rp on the critical current density can be disregarded for strong pins [ 16], we shall hereafter omit Rp in eq. (2.7a). We have considered the case of u >> Ujo, because the aim of this section is to investigate the behavior of u (r, t) in the presence of a thermal fluctuation force Rx(R, t). If we take into account the thermal fluctuation force Rx(r, t), then eq. (2.7a) is reduced to
~(Ou/Ot) =D2u-o~(u; Uso)u+Rx(r, t) .
(2.7b)
This type of equation is known as the Langevin equation[ 18 ] with the fluctuation characteristics of Rx given by
(RT (r, l) )s =0, (R-r (r', t' )'Rx (r+r', t+t' ) )s =2(kn TS(r)8(t) ,
(2.9a) (2.9b)
where ( )s represents a statistical average, ka the Boltzmann constant and T the absolute temperature. A usual process for carrying out the statistical average of eq. (2.7b) is to use the Fokker-Planck equation [ 22 ]. Since the averaging process is slightly complicated [ 16 ], a detailed process is shown in appendix A and we give here only the resulting equation:
((O/Ot) (u)~
= -o~s ( u ) ~ ;
(2.10)
o~s(t)=f ~V2Up ) + 6( ~V4Up) [U2o+ Sxx(O, t) + Syy(O, t) ] . In the above equations, Sxx(r, t) and fluctuation force, Rv, defined as
Svv(r, t)
(2.11)
are the correlation functions for pinned fluxoids for the thermal
S~j(r, t) = (ui(r' +r, t)ufir', t) )s; i,j=x, y.
(2.12)
The equations of Sirs are given in appendix A and the stationary solutions of these equations are given by
S,..,-(r)=kaTA,(r)exp[ _ \(~x2+y2 + ~ 4 4 ) l +2"2k a T A 2 ( r ) e x p [
{ x2+y2
z2
where AI and A2 are defined by eq. (A.5). The expression for S~y(r) can be obtained by substituting in the above expression for S~x(r), where the correlation lengths, l~, 144and/66, are given by
l~166
746
K. Yamafuji et al. I Thermally activatedflux creep
l,,=
' /4"=\a~J
,
/66
(2.14)
Suppose that a fluxoid at r = 0 suffers a unit displacement by a thermal fluctuation in the direction of 0 = ~/ 4 in the (x, y) plane, other fluxoids also suffer displacements due to correlations among pinned fluxoids. In fig. 1, the pattern o f fluxoid displacement in the plane of z = 0 is illustrated, and the attenuation o f the displacement propagation along the direction o f 0 = n / 4 is plotted in fig. 2. It can be seen from fig. 2 that the correlation among pinned fluxoids decreases sharply as e x p ( - p / 1 6 6 ) with p = (x 2 + y2)1/2, and hence the correlation length in the (x, y) plane for pinned fluxoids and for thermal fluctuations can be defined as/66. Thus it is reasonable to choose the average volume of a flux bundle characterizing a coherent motion of pinned fluxoids under thermal fluctuations RT as Vb = q ( 2166)2/44 ,
(2.15)
where ~/is a numerical factor of about 1. Note that the correlation length along fluxoids is/44, as can be seen from eq. (2.13). In eq. (2.15), 166 and/44 are defined by eq. (2.14), in which the modified Labusch parameter a s ( 0 ) is given b y e q . (2.11) as (x~(0) = / ~V2Up ) + 1 / ~V4Up )[U~o + S,.,(O, 0 ) + S,,,,(O, 0 ) ] , and the elastic moduli of fluxoid lattice, Ct ~ ,
C44
(2.16)
and C66, are given by [20,21]
Cil ~ C44 =B2/lzo,
(2.17a) B
C~6 -81XoK2 \
2
B
2 B2"~
< .17b)
2K-,/Bc2\
~| 0 .20
1.5
Y
FL
~,
g1.o. ~
,9 / /
+
0.5
/ /
f "
~
"t -
"I,
, "
Fig. l. A schematic illustration of the pattern of the pattern of fluxoid=displacements in the plane z=0, when a fluxoid at r=0 suffers a unit displacement by thermal fluctuation in the direction of the Lorentz force ( 0= ~/4). Other fluxoids also suffer displacements due to correlations among pinned fluxoids.
-0.3
i
-0.2
-0.1
0.1
0.2
t~
p
0.3 ( 1.1.m )
Fig. 2. Variation of correlation function, S~.x(0)+S,.,.(0), with p = (x 2+y2)~12. Since the correlation among pinned fluxoids decreases nearly as exp(-p/166), the correlation length in the (x, y) plane for thermal fluxtuations can be defined as/66.
K. Yamafujiet al. / Thermallyactivatedflux creep
747
In the above expressions, B is the flux density, #o is the magnetic permeability of vacuum, x is the GinzburgLandau parameter and Be2 is the upper critical flux density.
3. Thermal hopping rate of flux bundles Let us now discuss the thermally-activated hopping motion of flux bundles under infuence of the Lorentz force. For this purpose, it is convenient to choose the semi-macroscopic volume, in which flux density B can be regarded as constant, as an average volume Vb of each flux bundle, and to take into account the difference of B between successive flux bundles along the direction of the Lorentz force explicitly. Since the fluxoids inside each flux bundle move coherently, the systematic displacement of each fluxoid inside the concerning flux bundle can be described by eq. (2.6), in addition to the displacement due to the thermal fluctuation force Rx which also acts coherently on all the fluxoids inside the flux bundle. Then the average displacement of the concerning flux bundle, ub, obeys the following equation: (Vb(0Ub/Ot)
= -- (OUe/OUb)+ R v b ( t )
;
( OU~/OUb)= Vb[ ( ( ~V2Up ) +I ( ~ v4Up )u~,)Ub--FL ] ,
(3.1) (3.2)
where we have considered only one-dimensional displacement along the direction of the Lorentz force FL. These equations are applicable for the values of Ub between two successive maxima of Ue. The statistical characteristics of Rxb(t), the thermal fluctuation forces summed up over a flux bundle, are defined by (Rxb(t))s =0,
(3.3a)
(Rvb(t')Rvb(t+t')
) , =2~Vbks Td( t ) ,
(3.3b)
where ( )s represents the statistical average over many flux bundles located in a plane perpendicular to the direction of the Lorentz force. The probability distribution function P(Ub, t) for ub obeying eq. (3.1) can be described by the following Fokker-Planck equation [ 17 ]: OP
0 [Oue
.
~Vb--~=~UbL'-~Ub+ lCBT o--~b]P .
(3.4)
The Fokker-Planck equation of the above type is called the Smoluchowski equation [ 18 ]. The solution of this Smoluchowski equation in thermal equilibrium is given by 1 [ P~(Ub) = ~ exp
Ue (Ub) ] ~:~- j,
(3.5)
where Z is a normalization constant. The concerning process is, however, not in thermal equilibrium but is a thermally-activated motion of flux bundles in the presence of the Lorentz force. Thus the effective potential Uc of the concerning pinned fluxoidbundle is asymmetric for forward and backward motions along the direction of the Lorentz force. The m a x i m u m value of the first term on the right-hand-side of eq. (3.2.), ((ZV2U,,)+(1/ 6 ) ( ~,~74Up>/12 )Ub ' is the (semi-macroscopic) pinning-force density, Upbecause the flux flow occurs when the Lorentz-force density FL exceeds Fp, as can be seen from eq. (2.6). By differentiating this first term with respect to ub, we obtain:
Fo=JcB=oqdi
;
(3.6)
748
K. Yamafuji et al. / Thermally activated flux creep
O~L=I~V2Up1, di=I8(~,V2Up)/9(-7,V4Uo))l
'/2 ,
(3.7)
where O~Lis the Labusch parameter [ 14] and di the interaction distance [8 ]. The distance for which the force reaches the maximum in the summed-up pinning potential is given by do= (3/2)di, as shown in fig. 3a. Although a concrete expression for Fp as well as for eL and di depends on the species of pinning centers and their distribution, at least we can estimate F o, O~Land d~ experimentally by the so-called Campbell method [ 8,23 ], as shown in fig. 3b. Since d~ is usually much smaller than the fluxoid spacing ar [23], the peak of Ue(Ub) defined by eq. (3.2) in the forward direction along the Lorentz force is given by Uem - - C e ( 0 ) ~' ( F p - - I F L I
)do Vb/2,
(3.8a)
as can be seen from fig. 4, where the factor 2 comes from the integration of forces. It is to be noted that flux flow occurs when FL exceeds Fp, so that the summed-up potential shown in fig. 3a should be cut at ub = dp = (3/ 2)d,. Since the present expression for the summed-up pinning potential is rather formal, a detailed structure inside the pinning potential which gives a difference of the order of unity in eq. (3.8a) is disregarded. A detailed discussion on this point will be presented in section 6. Similarly, the peak of Ue in the backward direction is given by Uem - - U e ( 0 ) ~-- [FL [ a f J r ( F p - IFLI )dp V b / 2 ,
(3.8b)
where fir is the periodicity of Uc. Now let us discuss the thermally-activated hopping-rate of the concerned flux-bundle across the peak at ub~--dp. If we formally express the stationary solution of eq. (3.4) as
of
-L7
Uem
i 0
d~
Ub
Fig. 3. (a) Schematic illustration of the summed-up pinning-force potential. The width of the potential for which the pinning force becomes maximum is given by 2dp = 3d~, where d~ is the interaction distance [ 8 ]. (b) A schematic graph of the restoring force density F versus systematic displacement ub of fluxoids which be observed from the Campbell-method measurements [ 8,23 ]. The Labusch parameter aL and the interaction distance d~ are defined by the initial gradient and the displacement at which the extrapolation of the linear part reaches the critical state, respectively.
/~em
-af -(af-dp)
)
i
;
'
0
dp
)
,,
~f
.
ub
Fig. 4. Schematic illustration of the effective potential U¢ in the presence of the Lorentz force. In the forward direction along the Lorentz force, Ue reaches its maximum U~mat ub---dp. (see text. ) On the other hand, the maximum of Uem in the backward direction is located at ub ~- - [~f-dp].
K. Yamafujiet al. / Thermallyactivatedflux creep P(ub) =G(ub)Pe(Ub) ,
749 (3.9)
then G (Ub) is given by
u~
f (Vb Jb(u) du kBTPe(u~ '
G(ub) =
(3.10a)
lib
where Jb(u) is the forward flux of particles (i.e., flux bundles) across the peak of Uem, and Jb(U) is defined by
1 [-OUe kaT~__~b]p(ub)
(3.11)
Since [Pe(Ub)]-' has a sharp maximum at Ub~--dp, eq. (3.10a) is reduced to
G(Ub) ~--
exp
exp - 2 k ~ ( U b - d p
dUb
Ub \~---TkBTJ
Jb(dp)exp
.
(3.10b)
The fraction of flux-bundles staying inside the concerning potential-well but capable of proceeding in forward direction is given by do dp
~=
f P(ub)dub= f G(uu)Pe(ub)dUb ; 0
3
dp=~di.
(3.12a)
0
Inserting eq. (3.10b) into eq. (3.12a) and carrying out a similar calculation for Pe (Ub), which has a sharp maximum at Ub--~0, we obtain _, 27t [- Uem - Ue(O) l p~_7¢Jb(dp)exp[ ~j.
(3.12b)
Then the hopping frequency, f, of flux-bundles across the forward peak is given by
f= Jb(dp)/p= ~o~L exp
[
6"erak---Uo(O) ~ ]l "
(3.13a)
A similar calculation for the backward hopping motion of flux-bundles leads to
O/L
f = ~--~(exp
[ ~'~emSe (0)1] . ~-
(3.13b)
4. Creep rate of magnetic flux
For estimating the very slow variation of the flux density B with time due to thermally-activated flux creep, we can use a stationary hopping frequency, f or f, give by eqs. (3.13a) and (3.13b). Then the time-variation of B(x, t) is given by
Ot B(x, t)= [ B ( x - & , O f ( x - & , t)+B(x+gtf, Of(x+&, t ) - B ( x , t)f(x, t ) - B ( x , t)f(x, t)]
~- - ~[B(x, t)&(x, t){f(x, t)-f(x, t)}].
(4.1)
K. Yamafujiet al. / Thermallyactivatedflux creep
750
More generally, the above equation is reduced to the Maxwell equation:
OB/Ot =
-
V × E~r .
(4.2)
if we adopt the following Josephson relation [24 ] for the induced electric field Ecr due to flux creep:
Ecr=B×v~r :(J/J)Bfir~-f[1-exp(
IF~lfirVb~lexp[ - k B flJ T
(Fo--IFLI)dpVb]
(4.3)
where we have used eqs. (3.8a), (3.8b), (3.13a) and (3.13b). Explicit expressions for dp= (3/2)di and Vo are given by eqs. (3.7) and (2.15), respectively. For simplicity, let us confine the following discussion to a slab sample with the volume of ( L x X L y X L z ) with L, << L,. and L_-, and apply a magnetic field B in the z-direction and an electric current J in the y-direction. When the external magnetic field is increased (or decreased) and stopped at a certain value Be after cooling the sample in zero field, the field distribution inside the sample approaches quickly the so-called critical described by [ 8,19 ].
V x B , =#oJ¢(Bc) .
(4.4)
Since the flux distribution will keep the critical-state distribution, B = Be (x), if thermally-activated flux creep is disregarded, we can define the variation of B(x, t) due to flux creep as
B(x, t ) = B e ( x ) + b ( x , t ) .
(4.5)
Since b(x, t) is small compared with B~(x) for a strongly-pinning sample so far as the usual observable time interval is concerned, we only have to retain the first-order terms with respect to b. Then eqs. (4.2) and (4.3) are reduced to
Ob
~{Bc(X)Uc(X)exo[ fob
OJc.~gUcdi(Uc)Vb(Uc)q)"
- ~x-/~°-~c°) '
0t
- ~ B c ) ~aL(Bc) Vc(X)--ar( [1--exp(FL(Bc)(tf(Bc)
4---~; T-
Vb(B¢))]
ks T
_]J''
(4.6)
(4.7)
where we have used Fp = J~ (B) B and FL = J ( B ) B = B (OB/Ox)/IZo. In the actual situation, some traces of local flux flow still remain, even when the variation of B due to flux creep becomes dominant. Then the definition of the time 6 at which the flux creep is observed as dominant has some ambiguity, as well as the value of the flux-creep velocity v~ at t = 6. It can be proved easily that eq. (4.6) has a solution of the separated type, b(x, t) = bx(x)bt(t), and that the resulting expression for the solution is given by
B(x, t) =Bc(x) + fl(x)ln( t/t~) ,
(4.8a)
where fl(x) is defined as Be
[fl(x)[ =kaTJ~(Bc)
J~(B-~c(B) '
Uc(B~) = 3jc(Bc)Bcd~(Bc) Vb(Bc) .
(4.8b)
(4.9)
K. Yamafuji et al. / Thermally activated flux creep
751
The actual value of Vc can be obtained from the relation d(Bcv~)/dx= -Ifl(x)l/tc. When the external magnetic field is fixed at Be, the magnetization M per unit volume is given by M = ( B ) x - Be where ( )x represents the average with respect to x. Then the flux-creep rate is given by BO
i=lOM/Oln(t/tc)l=l(fl(X))xl=p---o-o~~
Bc
dBc Be
f
Be
dBj~(B)U~(B), ,
where Bo is the value of the B~ at x = 0 . If the external field Be is much larger than Bp, with Bp denoting the characteristic field at which the flux front reaches the center of slab, i.e., Bp~-I~oJ~(Bp)Lx/2, then eq. (4.10) is reduced to a simpler form. For example, if we estimate Jc(Be) from the observed magnetization curve at Be>>Bp by ]Me(Be) ] --~toJc(Be)Lx/4, then the averaged flux-creep rate between the two processes, where the external field is fixed at B~ after increasing and decreasing the external field, is given by
R/IMc I =kB T/Uc(Be)
(4.11 )
•
Whereas this expression is formally the same as those in existing theories [ 8,10-13 ], the detailed expression for U~ is different from the previous ones. Furthermore, U~ is a given function of observable quantities in this theory. We like to emphasize that the simple relation given by eq. (4.11 ) can be obtained only for the above-mentioned process. Otherwise, the apparent effective pinning potential Uo(Be) estimated from R/IMcl = k~ T/Uo (Be) shows some deviation from Uc(Be). A detailed discussion on this point is given in appendix B.
5. Comparison of the present theory with observed data The expression for Uc(B), which is essentially related to the relative magnetic-flux creep rate as shown in eq. (4.1 1 ), is given by eq. (4.9), in which the average volume of each flux bundle is typically given by eq. (2.15). In actual observations, however, we sometimes encounter a situation, which deviates from the assumptions made in the derivation of eq. (2.15). These situations are summarized in table I, and the corresponding expressions for Vb are also shown. In table I, the cases (c) and (d) usually appear for a thin film when a magnetic field is applied perpendicularly to the film-surface. A detailed discussion for these cases will be presented in the next section. Whereas the modified Labusch parameter as (0) in eq. (2.14 ) approaches zero as J--'Jc [ 16 ], as (0) is nearly equal to the Labusch parameter aL defined by eq. (3.7) at J
(5.1a)
Uc (B) ~- ]rl[J~(B)B] ,/2 [di(B) ]3/2 [af(B) ]2[C44 (B) ]1/2 for case ( b ) .
(5. lb)
Table I Expressions for the average volume of a flux bundle, Vb, in various cases. Case
(a)
(b)
(c)
(d)
144
144
144>Lz
144>L~
2/66 >> ar
2/66 ~ af
2/66 >> af
2/66 < ar
~ (2/66)2/44
~a2144
~ (2166)2L~
~arZL~
situation
Vb
K. Yamafujiet al. / Thermally activatedflux creep
752
it is well-known empirically that the following relation holds for the interaction distance, di [8,23 ]:
d,(B) = 6 a t ( B ) ,
(5.2)
where the fluxoid spacing af(B) is given by
af(B)
{ 2 (/)o) '/2 -if/
= \~
(5.3)
with ~o denoting the fluxoid quantum. The coefficient 6 does not depend on the magnetic field B but depends on the pinning strength and on the superconducting parameters. Insertion of eq. (5.2) into eqs. (5. l a) and (5. lb) leads to the B-dependance of Uc(B):
U¢(B) ~_3rl65/2 [J¢(B)B] -,/2 [af(B) ]5/2C66(B) [C44(B) ],/2
for case ( a ) ,
(5.4a)
Uc (B) ~- ~t163/2[Jc(B)B ] x/2[af(B) ]7/2 [C44(B) ],/2
for case (b) ,
(5.4b)
where the expressions for C44(B) and C66 (B) are given by eqs. (2.17a) and (2.17b), respectively. Detailed measurements of flux creep in conventional superconductors suitable for a quantitative comparison with the present theory are rather scarce, because detailed information on the pinning characteristics necessary for the quantitative comparison with the present theory is not specified in most of the existing measurements. Antesberger and Ullmaier [25 ] investigated hollow cylinders of N b - T a alloys containing normal-conducting precipitates. For the sample in which B~2=0.6 T and x = 4 . 5 at 4.2 K, for example, Fp is 3.1 × l07 N / m 3 at Be = 0.24 T. The observation of Uc(B~) was carried out by averaging the creep rates between increasing- and decreasing-field processes after zero-field cooling. Since eq. (4.11 ) can be applied to this process because Be is much larger than Bp( ~ 0.05 T), the observed result yields Uc(0.24 T ) = 0 . 2 3 eV. Since the observation of d~ was not carried out by Antesberger and Ullmaier, we adopt the value of di=af/20, observed in a similar sample of N b - T a [26]. With this value ofdi, we find that case (a) is realized in the range 0.1 < B / B c2<0.9. Since Uc for case (a) is given by eq. (5.4a), we compare the theoretical curve with the observed data for Uc(B~), and the result is shown in fig. 5. The quantitative agreement is quite satisfactory. Another detailed measurement of flux creep in conventional superconductors is provided by Beasley et al. [ 12 ] for Pb-10% T1. For the second sample in which Be2 = 0.2 T and ~c= 2.7 at 4.2 K, for example, Jc is 3.0 × 107 A / m 2 at Be=0.1 T. The observation of U¢(Be) was carried out by averaging the creep rates between increasingand decreasing-field processes after zero-field cooling. Because of Be >> Bp ~ 0.02 T, eq. (4. l 1 ) is applicable and
0.5 0.~
2 rnple 1
o
o
Sample 2
:~ 0.2 o.1
0
0.2
0.4
0.6
0.8
B/Bc2 Fig. 5. Comparison of the theoretical curve for Uc(B¢)given by eq. (5.4a) and the observed data for Nb-Ta studied by Antesbergerand Ullmaier [26 ]. We estimate Ucfrom eq. (5.4a) using the observed value of JcB.
0.4
0.6
0.8
B/Bcz
Fig. 6. Comparison of the theoretical curve for U¢(B~)obtained by eq. (5.4a) and the observed data for Pb-10%Tl given by Beasley et al. [12]. The values of d~ are chosen as af/9.94, af/1 1.7 and af/17.0 for sample 1, 2 and 3, respectively.
753
K. Yamafuji et al. / Thermally activatedflux creep A
60
lOkG
50
3kG 2kG
E ~t-4 0
2s g
~20
~I " '\
o
IkG N
c
~
~;' II
{D
O
= -25
0
i
10
i
10 2 Time ( s e c )
i
10a
0 C
',,7
~' -so
J -10
-5
5
10
Magnetic Fiel0 (k6)
Fig. 7. (a) Creep rates of the magnetizationfor a quench and melt growthprocessedsample in remanent states providedby Murakami et al. [9 ], where an external field was increased up to Bm and then decreased to zero. The values of Bm are indicated in the figure. (b) Magnetization curves for the melt-processedsample of Lx= 1.l mm observedby Murakami et al. [9 ]. the observed result yields Uc(0.1 T)~- 1.0 eV and (3/4)Vbdi ~- 5.0× 10 -26 m 4. Since the observation ofdi was not carried out carried out by Beasley et al., we estimated the value of di from the observed value of (3/ 4) Vbdi ~-5.0 × I 0-26 m 4. If we assume that case (a) is realized, the best result is given by di ~ af/11.7. With this value of di, we get 2/66~ 0.92 ~tm while ar is 0.15 ~tm, which confirms the assumed situation of 2/66 >> af for case (a), as expected. The comparison of eq. (5.4a) with observed data of Uc(B) by Beasley et al. [ 12] are shown in fig. 6. The agreement is also quite satisfactory. The assumed values of di in fig. 6 are somewhat larger than the observed value ofd~ = af/30 for Pb-20% TI provided by Matsushita et al. [27 ]. We note that a broad peak effect in J¢(B) is observed in the range of 0.5<~B/B¢2~0.9 for Beasley's sample, while no peak effect is observed in Matsushita's sample. Since it has been shown [28] that the di takes a larger value when a peak effect appears, the above-assumed values of d~ seem to be reasonable. Flux-creep measurements on a strongly-pinning bulk samples of the high-T~ oxide superconductors have only been carried out by Murakami et al. on quench and melt growth processed slabs of Y - B a - C u - O at 77 K [9 ]. Their measurements were carried out at the full-remanent state obtained by fixing Be at Be = 0 after Be was increased up to Bm> 0.4 T ~ 2Bp, and the results are shown as the uppermost curves in fig. 7a, which yield values of U0 of about 0.6 eV. Although this observed process did not satisfy the required situation leading to eq. (4.11 ), the value of Uc estimated with the relations shown in appendix B, using the observed magnetization shown in fig. 7b, is not very far from Uo i.e., Uc(0.2 T) ~0.75 eV. This type of melt processed sample, however, has a special configuration in which several tens of grains are nearly separated by normal-conducting bands [ 9 ]. Thus the local critical current density Jog inside the grains probably plays a dominant role for flux creep, as in a sintered sample. The typical values Jog and d~ observed by us, using Campbell's method [8,23 ], are J~g~ 2 × 10 ~° A / m a and di ~ a f / 1 0 at 0.2 T for similar samples of this type of melt-processed Y - B a - C u - O . Is we adopt these values, the value of/66 ~ (C66di/JcgB)l/2 is much smaller than af at 0.2 T, and hence the present situation may be classified into case (b), 2/66~
6. D i s c u s s i o n
According to the present theory, the relative creep rate of the magnetization is typically given by eq. (4.11 ), though one should realize that Uo(Be) estimated from the observed value of_Rby the relation of/~/JMc] = k a T /
754
K. Yamafuji et al. / Thermally activatedflux creep
Uo(Be) is not always equal to Uc(Be), as shown in appendix B. Whereas eq. (4.11 ) is formally the same as the expressions given in previous works [ 8,10-13 ], the expression for Ue(B) given by eq. (4.9):
Uc(B) = 3jc(B)Bdi(B) Vb(B)
(6. la)
is different from the previous expressions [8,10-13]. Furthermore, the present expression for Uc(B) is a function of observable quantities. That is, Vb(B) is given by eq. (2.15) or table I, where the correlation lengths of pinned fluxoids are given by 144 -----(C44/Ol'L)1/2= (C44di/JcB) 1/2,
(6.1b)
166 ,.~ (C66/OLL) 1/2= (C66di/JcB) 1/2,
(6. lc)
with the C44 and C66 shown in eq. (2.17a) and (2.17b). c~L or d~ can be determined by Campbell's method [8,23] from the relation Fp=J~B=aLdi. As shown in the previous section, the present expression for Uc(B) can explain quantitatively the existing observed data [9,12,25] by assuming reasonable values for the interaction distance, d~(Be). No observation of d~(B), however, has been carried out for the sample in which the creep rate of magnetization was observed [9,12,25 ], and hence a more detailed check_,on the applicability of the present theory is desirable. It is to be noted that the expressions for Uem and Uem given by eqs. (3.8a) and (3.8b) are not applicable for J in the y-direction. The electric field Ecr, induced by flux creep is given by eq. (4.3). Since we can put B
Vcr=LvBeaf(Be)fo
{
1-exp
[ 4Clf(Be)Uc(Be)j
)
exp
Uc(Be) ( 1 Jc(Be)}_J "~l, kaT
(6.2a)
where the thermal vibration frequency, fo (Be), of a flux bundle inside the effective pinning potential Ue is given by
aL Je(Be)pf(Be) f o - 2re(- 2nBedi(Be) "
(6.2b)
pf(Be) ~- (Be/Bc2)PN denotes the flux-glow resistivity, and we have used the relation (= B 2/pf(Be). The present expression for Vcr induced by the flux creep in thin film samples is somewhat different from the expressions presented in earlier works [3,8]. Furthermore, the present expressions forfo and Vcr are given functions of observable quantities including pinning parameters if we assume (tf(Be) ~-af(Be), as is usually done. It is to be noted [6] that, for estimating Uc(Be) from the observed data of Vcr using eq. (6.2a), all the quantities are functions of temperature except the external variables, Be and. It is, therefore, necessary to observe the T-dependences of Be2, x, J~ (Be) and di (Be). Furthermore, eq. (6.2a) cannot be applied in the region of < , as mentioned before.
K. Yamafuji et al. / Thermally activatedflux creep
755
The thermal vibration frequencyfo of flux bundles, given by eq. (6.2b), indicates that the motion of the flux bundles inside the effective pinning potential Ue, shown schematically in fig. 4, is a friction-damping motion triggered by thermal fluctuations. Since the value of 6 defined by di (Be) = ~ a f ( B e ) is typically of the order of ½n for high-x materials such as N b - T i [29] and melt-processed Y - B a - C u - O [9 ], 2nBedi(Be) ~ Bear(Be) is of the order of 10 - 7 m T at Be~ 5T. If we adopt the value o f p f ( 5 T ) ~ 1 0 - 6 ~ m for B i - S r - C a - C u - O [4] at 77 K, for example, )Cogiven by eq. (6.2b) leads to a value of the order of i 0Jc. Taking for Jc (5T) a value of 108~ 10'°A/ m 2 at 77 K, fo becomes of the order of 1 0 9 ~ l011 Hz, as suggested by Palstra et al. [4] and also by Malozemoff et al. [ 30 ]. A more detailed investigation is, however, necessary to check the applicability of the present theory. It is to be noted that the value of Uo ~ 0.6eV at 77 K for a quench and melt-growth processed bulk sample of Y - B a - C u - O [9] is not very small compared with the values of Uc observed in Pb-T1 and N b - T a at 4.2 K, as can be seen in figs. 5 and 6. Thus the main reason of the large flux-creep rates in high-To oxide superconductors observed at 77 K is the strong thermal activation at 77 K, as far as strongly-pinning bulk samples are concerned. While we have no observed data on the flux-creep rate in high-x conventional superconductors such as N b Ti at present, the value of C66 becomes smaller for strongly-pinning samples as can be seen from eq. (2.17b). Then case (b) in which 2/66 < af may be applicable, as in the melt-processed sample of Y - B a - C u - O [ 9 ]. In this case, eq. (5.4b) predicts that Uc increases with increasing Jc, although di=6af has some dependence on the pinning strength. Thus it is not unreasonable to expect a further increase of Uc for more strongly pinning samples of the high-To oxide superconductors. Glassy features due to random weakly coupled grains in high-temperature superconductors have been considered as another candidate for the mechanism of the magnetic relaxation [ 31 ]. However, agreement between the present theory and experimental data not only on new but also on conventional superconductors indicates that this magnetic relaxation can be systematically explained by the mechanism of thermally-activated flux creep. Recently, it was found [32] that the fluxoid lattice melts as liquid at high temperatures. The melting seems to appear at high temperatures where the irreversibility disappears [ 33 ]. Thus the region in which melting appears may be suppressed in a strongly-pinning sample. The tendency that the melting occurs more easily in B i - S r - C a - C u - O with smaller pinning forces than in Y - B a - C u - O seems to support this idea. However, the flux motion may be activated by the mechanical oscillations in the measurement of the fluxoid lattice melting [32]. This may lead to a difference from the usual flux creep due to thermal activation only.
Acknowledgment The authors would like to express their sincere appreciation to Dr. Masato Murakami and the research group in Nippon Steel Co. for showing their observed data before publication.
Appendix A The probability function P(u, t) for the relative displacement u obeying the Langevin equation (2.7b) is described by the following Fokker-Planck equation [22 ]:
~-i=~Jdr~u'L6u
kB
P,
(A.1)
where V is the volume of the semi-macroscopic region in which B can be regarded as nearly constant and ~b is the potential functional leading to the right-hand-side ofeq. (2.7b) by taking the variation of ~b with respect to u:
756
K. Yamafuji et al. I Thermally activatedflux creep
1 "+'C44
1
1
~
+
~
4
4+ 1
+(~y)
-[-C66\ ay 3t" a x J ) + ( C 1 1 - 2 C 6 6 )
1
~Y) .
(A.2)
Carrying out a statistical average of u using eq. (A. 1 ), we obtain eq. (2.10). Similarly, a statistical average of ux(r' +r, t)Ux(r', t) leads to the equation for the correlation function, S~x(r, t) [ 16]: O~
( 2
r
02
02
0 2 "]
Sxx(r, t ) = - a ~ ( t ) S , ~ ( r , t)+ICII~..2+C667.z+C44-~_21S~Ar, t) " k ox oy oz _1
+ (C~ -C66) ~
02
(A.3)
Sxv(r, t) + k B T r ( r ) .
We can get similar equations for S~y(r, t) and Sxy(r, t). The Fourier transform of this equation leads to the spectrum of the correlation function: k2 2 /~.,-.,'(k) ~'~"k B T~ 2~-
2
2
1
kx+k~ C,,(k~+k~)+C4~k~+o~(0)
kk ~
1
1
k~+k~ C66(k~+k~)+C44k~+OLs(0) '
(A.4)
Similar spectra for Z~y and X~ can be obtained [ 14,16 ]. Carrying out the inverse Fourier transform of Xx.~,we get eq. (2.13), where A~ (r), A2 (r) are given respectively by
1 (
(x2+y 2 Z 2"~ , ~ 2l 2, "~ A 1(r) --8~Cll/4 4 _ (1 +cos20) \ / - - - - ~ +/--~-4) *c°SZVx2-T~Y2,] ' Ae(r)=.8~6144
(X2+y 2 Z2"~ ( l - c o s 2 0 ) \ 126 +
..
(A.5a)
2/26
)-coszv ),
(A.Sb)
with 0=tan -l ( y / x ) .
Appendix B For a strongly-pinning bulk sample of a conventional superconductor, the magnetic-field dependence of the critical current density, Jc(B), can be approximated by [34 ] Jc(B) =Jc(Bp) ( B / B o ) r - ' [ 1 - (B/Bc2) 1'/[ 1 - (Bo/Bce) ] ' ,
(B. la)
where ? and e are numerical pinning parameters taking a value in the ranges of 0 < 7-< 1 and 1 -< E< 2, respectively, and Bp is the characteristic field at which the flux reaches the center of the sample. If we confine the present discussion to the range of B of B¢~ << Bp < B << Be> for simplicity, eq. (B. 1a) is reduced to J A B ) = J A & ) (B/Bp) > ' •
(B.lb)
For the case of a slab considered in section 4, therefore, eq. (4.4) is reduced to 052-:702= v,
(B.2)
where v is the sign factor taking the value of +_1 according to the direction of the Lorentz force, and 2 and /~ are normalized quantities: 2=2x/L,-,
~=B/Bp;
Bp=#oJc(Bp)Lx/2.
(B.3)
K. Yamafujiet al. / Thermallyactivatedflux creep
757
With the aid of these normalized quantities, eq. (4.10) is reduced to
ka T
bo
bc
Ue(Bp )
R--Uc~p)BP f d b c f d~l-TVc(bap ) ' he be
(B.4)
where we have used eqs. (B.lb) and (B.3). Similarly, eqs. (5.3a) and (5.3b) lead to
Uc(bBp) B ~ ( 3 _ 2 7 ) / 4 Ue(Bp )
for case (a) ,
(B.5a)
Ue(bBo) -~ 5(27_3)/4
for case (b) .
(B.5b)
Uc(Bp)
Then eq. (B.4) yields
R-
kBT B 4(2-7)(5-2Yb(e9-27)/4-b(~S-2Y)/4bo+ Ue(Bo) P 5 - 2 y \ 9 - 2 7
4---~b~9-2y)/4~ 9-27
u
]
R - kBT B 4(2-7)13+27b~7+z~,/4_b~3+2~)/4bo+ a .__£~b(7+2,~,/4 S -U~(Bp) p 3 + 2 y \ 7 + 2 y 7+2y e ]
forcase (a),
forcase (b)
(B.6a)
(B.6b)
•
On the other hand, the magnetization in the critical state is given by
Mc= ( Bc(x) ).~-Be =BpI 23~ ( b3-y-~3o-Y)-be l ,
(B.7)
where/~o is defined as
b?~-~'=b~-~- v.
(B.8)
Then the creep rate of the magnetization is given by
R/IMcI=kBT/Uo(Be); where
g(be)
Uo(Be)=Uc(Bp)/g(be),
(B.9)
is defined as
g ( b e ) = - - -1- 4(2--y)(5--2Yb(9_2y)/4_h(5_2y)/4/h2_y [m[ 5 - 2 7 \ 9 - 2 7 e ~e ,~e
l))l/(2_y)..[_ 4__.~(b2_7,_v)(9_2y)/4(2_~,)"~ --
9-2y
]
for case (a) ,
g ( b e ) - -Iml -1- -
4(2-y).(3+2Yb(7+2~)/4_].l(3+2y)/4(h2_~ 3+29, \ 7 + 2 y ~e ,~e --
(B.10a)
v)l/~2_v>+7+_~7)(b~_y -
p)(7+2?)/4(2_~,))
for case (b) ,
(B.10b)
with the definition of m:
m=23~[b 3-y- (b(e2-y- v)(3-y)/(2-y) ] - b e . For example, when Be is stopped at Bp in an increasing process of v = + l,
(B.11)
g(be)
is given by
758
K. Yamafuji et al. / Thermally activated flux creep
g~(1)-
4(2-7)(3-7) 9-27
for case (a) ,
(B.12a)
gi(1)-
4(2-7)(3-7) 7+27)
for case (b) .
(B.12b)
On the other hand, when Be is stopped at zero in a decreasing process o f v = - 1 (i.e., for the remanent state), we obtain 16(3-7) go(0) = ( 5 _ 2 7 ) ( 9 _ 2 7 ) g0(0)=
16(3-7) (5-27)(9-27)
for case (a)
(B.13a)
for case ( b ) .
(B.13b)
For practical conventional superconductors such as N b - T i and Nb3Sn, it has been known that the value o f 7 is nearly equal to ½ [ 35 ]. For this case of 7 = ½, both eqs. (B. 12a) and (B. 12b) lead to g~( 1 ) = ~-~ so that we have Uc(Bp)=~Uo(Bp). Both eqs. (B.13a) and (B.13b) lead to g o ( 0 ) = ~ for 7=½, so that we get Uc(0) = ~ Uo(0) for the full remanent state. Since the magnetization curves shown in fig. 7b are quite similar to those for 7= ½, we can use this ratio of Uc(O)/Uo(O)=~. It is to be noted that the relation of Uc(Bp) to Uo(B¢)becomes more complicated in the range of B<~Bo2, though we are able to derive the required relation with the present scheme.
References [ 1 ] Y. Yeshurun and A.P. Malozemoff, Phys. Rev. Lett. 60 (1988) 2202. [ 2 ] M. Tuominen, A.M. Goldman and M.L. Mecartney, Phys. Rev. B 37 ( 1988 ) 548. [3] M. Tinkham, Phys. Rev. Len. 61 (1988) 1658. [4] T.T.M. Palstra, B. Batlogg, L.F. Schneemeyer and J.V. Waszczak, Phys. Rev. Lett. 61 (1988) 1662. [5] M.E. McHenry, M.P. Meley, E.L. Venturini and D.L Ginley, Phys. Rev. B 39 (1989) 4784. [6 ] K. Enpuku, T. Kisu, R. Sako, K. Yoshida, M. Takeo and K. Yamafuji, submitted to Jpn. J. Appl. Phys. [ 7 ] D. Dimos, P. Chaudhari, J. Mannhart and F.K. LeGoues, Phys. Rev. Lett. 61 ( 1988 ) 219. [8] A.M. Campbell and J.E. Evens, Adv. Phys, 21 (1972) 199. [9 ] M. Murakami, M. Morita and N. Koyama, submitted to Jpn. J. Appl. Phys. [ 10] P.W. Anderson, Phys. Rev. Lett. 9 (1962) 309. [ l 1 ] P.W. Anderson and Y.B. Kim, Rev. Mod. Phys. 36 (1964) 39. [ 12] M.R. Beasley, R. Labusch and W.W. Webb, Phys. Rev. 181 (1969) 682. [ 13 ] C.W. Hagen, R.P. Griessen and E. Salomons, Physica C 157 ( 1989 ) 199. [ 14] R. Labusch, Crystal Lattice Defects 1 (1969) 1. [ 15 ] A.I. Larkin and Yu.N. Ovchinnikov, J. Low Temp. Phys. 34 ( 1979 ) 409. [ 16] K. Yamafuji, T. Matsushita, T. Fujiyoshi and K. Toko, Adv. Cryog. Eng. Mater. 34 (1988) 707. [ 17] H.A. Kramers, Physica 7 (1940) 284. [ 18 ] S. Chandrasekhar, Rev. Mod. Phys. 15 ( 1943 ) 1. [ 19] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [20] R. labusch, Phys. Star. Sol. (b) 32 (1969) 439. [21 ] E.H. Brandt, Phys. Rev. B 34 (1986) 6514. [22] C. Murakami and H. Tomita, Prog. Theor. Phys. 60 (1978) 683. [23] A.M. Campbell, J. Phys. C 2 (1969) 1492. [24] B.D. Josephson, Phys. Lett. 16 (1965) 242. [ 25 ] G. Antesberger and H. Ullmaier, Philos. Mag. 29 ( 1974 ) 1101. [26 ] T. Matsushita, T. Honda, Y. Hasegawa and K. Yamafuji, Mem. Fac. Eng. Kyushu Univ. 43 (1983) 233. [27] T. Matsushita, T. Tanaka and K. Yamafuji, J. Phys. Soc. Jpn. 46 (1979) 756. [28] T. Matsushita and T. Tanaka, J. Phys. Soc. Jpn. 47 (1979) 1433.
K. Yamafuji et al. / Thermally activated flux creep [29] T. Matsushita and H. Kiipfer, J. Appl. Phys. 63 (1988) 5048. [ 30 ] A.P. Malozemoff, T.K. Worthington, Y. Yeshurun, F. Holtzberg and P.H. Kes, Phys. Rev. B 38 ( 1988 ) 7203. [31 ] A.C. Mota, A. Pollini, P. Visani, K.A. Muller and J.G. Bednorz, Phys. Rev. B 36 (1987) 4011. [ 32 ] A. Khurana, Phys. Today 42 ( 3 ) ( 1989 ) 17. [33] P.L. Gammel, L.F. Schneemeyer, J.V. Waszczak and D.J. Bishop, Phys. Rev. Lett. 61 (1988) 1666. [34] F. Irie and K. Yarnafuji, J. Phys. Soc. Jpn. 23 (1967) 255. [35] M. Noda, K. Funaki and K. Yamafuji, Mem. Fac. Eng. Kyushu Univ. 46 (1986) 63.
759
A T H E O R Y O F T H E R M A L L Y A C T I V A T E D F L U X C R E E P I N N O N I D E A L T Y P E II SUPERCONDUCTORS
K. YAMAFUJI, T. FUJIYOSHI, K. T O K O and T. M A T S U S H I T A Department of Electronics, Kyushu University 36 Fukuoka 812, Japan Received 26 may 1989
The relaxation of the magnetization due to thermally activated flux creep in a bulk sample of a nonideal type II superconductor is discussed theoretically. Firstly, an expression for the average volume of a flux bundle characterizing its coherent thermal hopping motion is derived by starting with an equation for the local force balance on pinned fluxoids. Secondly, a thermal hopping rate of flux bundles is derived from the Fokker-Planck equation of pinned flux bundles. These results are used to derive expressions for both the time dependence of the magnetization and the induced electric field due to flux creep. It is shown that the present theory can quantitatively explain the existing observed data of the flux-creep rate, not only in conventional superconductors such as Nb-Ta and Pb-TI but also in strongly-pinning bulk samples of the high-Tooxide superconductors such as Y-Ba-Cu-O.
1. Introduction It has been p o i n t e d out that t h e r m a l l y - a c t i v a t e d flux-creep rates in high-To oxide superconductors are rem a r k a b l y larger than those in conventional superconductors, especially at higher t e m p e r a t u r e s [ 1-6 ]. This has been a t t r i b u t e d to the following sources: i) thermal fluctuation is stronger at higher t e m p e r a t u r e s a n d ii) pinning forces in existing high-To oxide superconductors are much weaker than those in conventional superconductors p r e s u m a b l y because o f the very short coherence lengths o f the high-To superconductors [ 1,7 ]. In conventional nonideal type II superconductors, the effects o f thermally-activated flux creep on their practical applications have been considered as not very serious a r o u n d 4.2 K [ 8 ]. While a recently observed fluxcreep rate at 77 K in a fairly strongly pinning bulk sample o f Y - B a - C u - O [9 ] is very small c o m p a r e d to the flux-creep rate in the so-called single crystals o f the high-T~ oxide superconductors, it is still possible that the effects o f t h e r m a l l y - a c t i v a t e d flux creep should be taken into account in detail when designing superconducting devices using high-Tc oxide superconductors. Whereas several theories have been presented on thermally-act i v a t e d flux creep in nonideal type II superconductors [ 8 , 1 0 - I 3], none o f these theories can explain quantitatively the existing o b s e r v e d d a t a even in conventional superconductors. It is, therefore, desirable to present a quantitative theory o f t h e r m a l l y - a c t i v a t e d flux creep in bulk samples o f conventional superconductors as well as o f high-T~ oxide superconductors, at least for strongly-pinning bulk samples which m a y be usable for practical applications. In this paper, we start with so-called Labusch equation [ 14 ] which describes the local force-balance on pinned fluxoids and which has been used to derive expressions for the semi-macroscopic pinning force density, Fp = B X J~ [ 14-16 ]. After m a k i n g a slight m o d i f i c a t i o n o f this equation so as to describe the local Lorentz-force density exactly, we derive expressions for correlation lengths o f p i n n e d fluxoids u n d e r thermal fluctuation forces. Since this sort o f correlated v o l u m e o f p i n n e d fluxoids gives an average v o l u m e o f each flux bundle characterizing its coherent thermal h o p p i n g motion, we derive a local force-balance equation o f flux bundles by integrating the starting equation over the average v o l u m e o f a flux bundle. This type o f the F o k k e r - P l a n c k equation [ 17 ] d e r i v e d from this local force-balance equation is known as the Smoluchowski equation [ 18 ], a n d has been used 0 9 2 1 - 4 5 3 4 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
744
K. Yamafuji et aL / Thermally activatedflux creep
to derive the thermal hopping rate of particles across potential peaks. The resulting expression for the thermal hopping rate of flux bundles across the summed-up pinning potential is used to derive an equation describing the semi-macroscopic flux density, B(r, t). The combination of this equation and the critical-state model [8,19 ] leads to a final expression for the thermally-activated flux-creep rate in a bulk sample of nonideal type II superconductors. Since the present expression for the fux-creep rate is a given function of observable physical quantities only, the present expression can be compared quantitatively with observed data. It is shown that the present theory can explain the existing observed data quantitatively, as far as fairly strongly-pinning bulk samples including high-Tc oxide superconductors are concerned.
2. Average volume of flux bundles Let us approximate the periodic fluxoid lattice in the absence of pins as a continuum with an elastic modulus tensor 1~ [20,21 ]. If pinning centers are introduced at the positions r=rn ( n = 1, 2, ... ,Np) inside the bulk sample, a volume element of the fluxoid continuum at the position r suffers a displacement uo(r). Since the change of ~ due to the introduction of pinning centers is usually negligible, the equation representing the local force-density balance can be expressed as
[)2'Uo- ~ VU,(r+uo-r,,)=O, pt
(2.1)
where U p ( r - r n ) is the elementary pinning potential and D2 is defined as [14]
02 02 a2 CII ~ -J-C66 ~ q- C44 OZ-'~" D2
02 (C11 -- C66)axay
02 02 02 C66~x2 ~-Cll--0y2 Jr-C44 ~z2
02 (Cii--C66)
(2.2)
In eqs. (2.1) and (2.2), the summation with respect to n is taken over a unit volume and the z-axis is chosen along the direction of the fluxoids. Next, let us apply a local transport current and let J represent the transport current density averaged over a semi-macroscopic volume in which the flux density B can be regarded as nearly constant with respect to r. Then the displacement uj(r, t) of the volume element at the position r at time t caused by the presence of J obeys the local force-density balance given by ((OUg/Ot) =D2" (Ug +Uo) - ~ V U v ( r + u o + u g - r n ) ,
(2.3)
n
where ( is the friction coefficient of fiuxoids per unit volume. The semi-macroscopic Lorentz-force density, F L = J × B , can be defined as the mean distorting force density obtained by averaging if2 "Us over the above-mentioned semi-macroscopic volume. Then we can formally express F L by FL = ( ~)2 "Uj ) = D2 "Ujo ,
(2.4)
where ( ) denotes the averaging over the above-mentioned semi-macroscopic volume, and Ujo(r, t) is a systematic displacement of fluxoids without shear induced by the application of the transport current. Then we can define the relative displacement u(r, t) as a deviation of uj from Ujo by
K. Yamafuji et al. / Thermally activatedflux creep
u=us-ujo
;
(u)=0.
745 (2.5)
When J is smaller than the so-called critical current density Jc, no flux flow occurs so that the magnitude of uj does not vary appreciably with time. If we confine the present discussion to the case of usual pinning strength in which no large plastic deformation of the fluxoid lattice is induced by an introduction of pinning centers, Up in eq. (2.3) can be expanded with respect to us. Then the average of eq. (2.3) over the above-mentioned semi-macroscopic volume around the position r leads to
~( OUjo/Ot) =FL -- ( ( E V2UpI -Jr~l ~ V4Uplu2o )UJO.
(2.6)
Insertion of eq. ( 2 . 6 ) i n t o eq. ( 2 . 3 ) y i e l d s '
((Ou/Ot) = 0 2 . u - a ( u ; Ujo)u+R o (r,
t) ;
(2.7a)
Og(/,/~/,/j0)~ ( E V2Up/ (..~1 E v 4 g p / (u20 "9ffu2) '
(2.8)
where Rp(r, t) is a remaining "fluctuation" pinning force defined by ( R p ) =0. Since the effect of Rp on the critical current density can be disregarded for strong pins [ 16], we shall hereafter omit Rp in eq. (2.7a). We have considered the case of u >> Ujo, because the aim of this section is to investigate the behavior of u (r, t) in the presence of a thermal fluctuation force Rx(R, t). If we take into account the thermal fluctuation force Rx(r, t), then eq. (2.7a) is reduced to
~(Ou/Ot) =D2u-o~(u; Uso)u+Rx(r, t) .
(2.7b)
This type of equation is known as the Langevin equation[ 18 ] with the fluctuation characteristics of Rx given by
(RT (r, l) )s =0, (R-r (r', t' )'Rx (r+r', t+t' ) )s =2(kn TS(r)8(t) ,
(2.9a) (2.9b)
where ( )s represents a statistical average, ka the Boltzmann constant and T the absolute temperature. A usual process for carrying out the statistical average of eq. (2.7b) is to use the Fokker-Planck equation [ 22 ]. Since the averaging process is slightly complicated [ 16 ], a detailed process is shown in appendix A and we give here only the resulting equation:
((O/Ot) (u)~
= -o~s ( u ) ~ ;
(2.10)
o~s(t)=f ~V2Up ) + 6( ~V4Up) [U2o+ Sxx(O, t) + Syy(O, t) ] . In the above equations, Sxx(r, t) and fluctuation force, Rv, defined as
Svv(r, t)
(2.11)
are the correlation functions for pinned fluxoids for the thermal
S~j(r, t) = (ui(r' +r, t)ufir', t) )s; i,j=x, y.
(2.12)
The equations of Sirs are given in appendix A and the stationary solutions of these equations are given by
S,..,-(r)=kaTA,(r)exp[ _ \(~x2+y2 + ~ 4 4 ) l +2"2k a T A 2 ( r ) e x p [
{ x2+y2
z2
where AI and A2 are defined by eq. (A.5). The expression for S~y(r) can be obtained by substituting in the above expression for S~x(r), where the correlation lengths, l~, 144and/66, are given by
l~166
746
K. Yamafuji et al. I Thermally activatedflux creep
l,,=
' /4"=\a~J
,
/66
(2.14)
Suppose that a fluxoid at r = 0 suffers a unit displacement by a thermal fluctuation in the direction of 0 = ~/ 4 in the (x, y) plane, other fluxoids also suffer displacements due to correlations among pinned fluxoids. In fig. 1, the pattern o f fluxoid displacement in the plane of z = 0 is illustrated, and the attenuation o f the displacement propagation along the direction o f 0 = n / 4 is plotted in fig. 2. It can be seen from fig. 2 that the correlation among pinned fluxoids decreases sharply as e x p ( - p / 1 6 6 ) with p = (x 2 + y2)1/2, and hence the correlation length in the (x, y) plane for pinned fluxoids and for thermal fluctuations can be defined as/66. Thus it is reasonable to choose the average volume of a flux bundle characterizing a coherent motion of pinned fluxoids under thermal fluctuations RT as Vb = q ( 2166)2/44 ,
(2.15)
where ~/is a numerical factor of about 1. Note that the correlation length along fluxoids is/44, as can be seen from eq. (2.13). In eq. (2.15), 166 and/44 are defined by eq. (2.14), in which the modified Labusch parameter a s ( 0 ) is given b y e q . (2.11) as (x~(0) = / ~V2Up ) + 1 / ~V4Up )[U~o + S,.,(O, 0 ) + S,,,,(O, 0 ) ] , and the elastic moduli of fluxoid lattice, Ct ~ ,
C44
(2.16)
and C66, are given by [20,21]
Cil ~ C44 =B2/lzo,
(2.17a) B
C~6 -81XoK2 \
2
B
2 B2"~
< .17b)
2K-,/Bc2\
~| 0 .20
1.5
Y
FL
~,
g1.o. ~
,9 / /
+
0.5
/ /
f "
~
"t -
"I,
, "
Fig. l. A schematic illustration of the pattern of the pattern of fluxoid=displacements in the plane z=0, when a fluxoid at r=0 suffers a unit displacement by thermal fluctuation in the direction of the Lorentz force ( 0= ~/4). Other fluxoids also suffer displacements due to correlations among pinned fluxoids.
-0.3
i
-0.2
-0.1
0.1
0.2
t~
p
0.3 ( 1.1.m )
Fig. 2. Variation of correlation function, S~.x(0)+S,.,.(0), with p = (x 2+y2)~12. Since the correlation among pinned fluxoids decreases nearly as exp(-p/166), the correlation length in the (x, y) plane for thermal fluxtuations can be defined as/66.
K. Yamafujiet al. / Thermallyactivatedflux creep
747
In the above expressions, B is the flux density, #o is the magnetic permeability of vacuum, x is the GinzburgLandau parameter and Be2 is the upper critical flux density.
3. Thermal hopping rate of flux bundles Let us now discuss the thermally-activated hopping motion of flux bundles under infuence of the Lorentz force. For this purpose, it is convenient to choose the semi-macroscopic volume, in which flux density B can be regarded as constant, as an average volume Vb of each flux bundle, and to take into account the difference of B between successive flux bundles along the direction of the Lorentz force explicitly. Since the fluxoids inside each flux bundle move coherently, the systematic displacement of each fluxoid inside the concerning flux bundle can be described by eq. (2.6), in addition to the displacement due to the thermal fluctuation force Rx which also acts coherently on all the fluxoids inside the flux bundle. Then the average displacement of the concerning flux bundle, ub, obeys the following equation: (Vb(0Ub/Ot)
= -- (OUe/OUb)+ R v b ( t )
;
( OU~/OUb)= Vb[ ( ( ~V2Up ) +I ( ~ v4Up )u~,)Ub--FL ] ,
(3.1) (3.2)
where we have considered only one-dimensional displacement along the direction of the Lorentz force FL. These equations are applicable for the values of Ub between two successive maxima of Ue. The statistical characteristics of Rxb(t), the thermal fluctuation forces summed up over a flux bundle, are defined by (Rxb(t))s =0,
(3.3a)
(Rvb(t')Rvb(t+t')
) , =2~Vbks Td( t ) ,
(3.3b)
where ( )s represents the statistical average over many flux bundles located in a plane perpendicular to the direction of the Lorentz force. The probability distribution function P(Ub, t) for ub obeying eq. (3.1) can be described by the following Fokker-Planck equation [ 17 ]: OP
0 [Oue
.
~Vb--~=~UbL'-~Ub+ lCBT o--~b]P .
(3.4)
The Fokker-Planck equation of the above type is called the Smoluchowski equation [ 18 ]. The solution of this Smoluchowski equation in thermal equilibrium is given by 1 [ P~(Ub) = ~ exp
Ue (Ub) ] ~:~- j,
(3.5)
where Z is a normalization constant. The concerning process is, however, not in thermal equilibrium but is a thermally-activated motion of flux bundles in the presence of the Lorentz force. Thus the effective potential Uc of the concerning pinned fluxoidbundle is asymmetric for forward and backward motions along the direction of the Lorentz force. The m a x i m u m value of the first term on the right-hand-side of eq. (3.2.), ((ZV2U,,)+(1/ 6 ) ( ~,~74Up>/12 )Ub ' is the (semi-macroscopic) pinning-force density, Upbecause the flux flow occurs when the Lorentz-force density FL exceeds Fp, as can be seen from eq. (2.6). By differentiating this first term with respect to ub, we obtain:
Fo=JcB=oqdi
;
(3.6)
748
K. Yamafuji et al. / Thermally activated flux creep
O~L=I~V2Up1, di=I8(~,V2Up)/9(-7,V4Uo))l
'/2 ,
(3.7)
where O~Lis the Labusch parameter [ 14] and di the interaction distance [8 ]. The distance for which the force reaches the maximum in the summed-up pinning potential is given by do= (3/2)di, as shown in fig. 3a. Although a concrete expression for Fp as well as for eL and di depends on the species of pinning centers and their distribution, at least we can estimate F o, O~Land d~ experimentally by the so-called Campbell method [ 8,23 ], as shown in fig. 3b. Since d~ is usually much smaller than the fluxoid spacing ar [23], the peak of Ue(Ub) defined by eq. (3.2) in the forward direction along the Lorentz force is given by Uem - - C e ( 0 ) ~' ( F p - - I F L I
)do Vb/2,
(3.8a)
as can be seen from fig. 4, where the factor 2 comes from the integration of forces. It is to be noted that flux flow occurs when FL exceeds Fp, so that the summed-up potential shown in fig. 3a should be cut at ub = dp = (3/ 2)d,. Since the present expression for the summed-up pinning potential is rather formal, a detailed structure inside the pinning potential which gives a difference of the order of unity in eq. (3.8a) is disregarded. A detailed discussion on this point will be presented in section 6. Similarly, the peak of Ue in the backward direction is given by Uem - - U e ( 0 ) ~-- [FL [ a f J r ( F p - IFLI )dp V b / 2 ,
(3.8b)
where fir is the periodicity of Uc. Now let us discuss the thermally-activated hopping-rate of the concerned flux-bundle across the peak at ub~--dp. If we formally express the stationary solution of eq. (3.4) as
of
-L7
Uem
i 0
d~
Ub
Fig. 3. (a) Schematic illustration of the summed-up pinning-force potential. The width of the potential for which the pinning force becomes maximum is given by 2dp = 3d~, where d~ is the interaction distance [ 8 ]. (b) A schematic graph of the restoring force density F versus systematic displacement ub of fluxoids which be observed from the Campbell-method measurements [ 8,23 ]. The Labusch parameter aL and the interaction distance d~ are defined by the initial gradient and the displacement at which the extrapolation of the linear part reaches the critical state, respectively.
/~em
-af -(af-dp)
)
i
;
'
0
dp
)
,,
~f
.
ub
Fig. 4. Schematic illustration of the effective potential U¢ in the presence of the Lorentz force. In the forward direction along the Lorentz force, Ue reaches its maximum U~mat ub---dp. (see text. ) On the other hand, the maximum of Uem in the backward direction is located at ub ~- - [~f-dp].
K. Yamafujiet al. / Thermallyactivatedflux creep P(ub) =G(ub)Pe(Ub) ,
749 (3.9)
then G (Ub) is given by
u~
f (Vb Jb(u) du kBTPe(u~ '
G(ub) =
(3.10a)
lib
where Jb(u) is the forward flux of particles (i.e., flux bundles) across the peak of Uem, and Jb(U) is defined by
1 [-OUe kaT~__~b]p(ub)
(3.11)
Since [Pe(Ub)]-' has a sharp maximum at Ub~--dp, eq. (3.10a) is reduced to
G(Ub) ~--
exp
exp - 2 k ~ ( U b - d p
dUb
Ub \~---TkBTJ
Jb(dp)exp
.
(3.10b)
The fraction of flux-bundles staying inside the concerning potential-well but capable of proceeding in forward direction is given by do dp
~=
f P(ub)dub= f G(uu)Pe(ub)dUb ; 0
3
dp=~di.
(3.12a)
0
Inserting eq. (3.10b) into eq. (3.12a) and carrying out a similar calculation for Pe (Ub), which has a sharp maximum at Ub--~0, we obtain _, 27t [- Uem - Ue(O) l p~_7¢Jb(dp)exp[ ~j.
(3.12b)
Then the hopping frequency, f, of flux-bundles across the forward peak is given by
f= Jb(dp)/p= ~o~L exp
[
6"erak---Uo(O) ~ ]l "
(3.13a)
A similar calculation for the backward hopping motion of flux-bundles leads to
O/L
f = ~--~(exp
[ ~'~emSe (0)1] . ~-
(3.13b)
4. Creep rate of magnetic flux
For estimating the very slow variation of the flux density B with time due to thermally-activated flux creep, we can use a stationary hopping frequency, f or f, give by eqs. (3.13a) and (3.13b). Then the time-variation of B(x, t) is given by
Ot B(x, t)= [ B ( x - & , O f ( x - & , t)+B(x+gtf, Of(x+&, t ) - B ( x , t)f(x, t ) - B ( x , t)f(x, t)]
~- - ~[B(x, t)&(x, t){f(x, t)-f(x, t)}].
(4.1)
K. Yamafujiet al. / Thermallyactivatedflux creep
750
More generally, the above equation is reduced to the Maxwell equation:
OB/Ot =
-
V × E~r .
(4.2)
if we adopt the following Josephson relation [24 ] for the induced electric field Ecr due to flux creep:
Ecr=B×v~r :(J/J)Bfir~-f[1-exp(
IF~lfirVb~lexp[ - k B flJ T
(Fo--IFLI)dpVb]
(4.3)
where we have used eqs. (3.8a), (3.8b), (3.13a) and (3.13b). Explicit expressions for dp= (3/2)di and Vo are given by eqs. (3.7) and (2.15), respectively. For simplicity, let us confine the following discussion to a slab sample with the volume of ( L x X L y X L z ) with L, << L,. and L_-, and apply a magnetic field B in the z-direction and an electric current J in the y-direction. When the external magnetic field is increased (or decreased) and stopped at a certain value Be after cooling the sample in zero field, the field distribution inside the sample approaches quickly the so-called critical described by [ 8,19 ].
V x B , =#oJ¢(Bc) .
(4.4)
Since the flux distribution will keep the critical-state distribution, B = Be (x), if thermally-activated flux creep is disregarded, we can define the variation of B(x, t) due to flux creep as
B(x, t ) = B e ( x ) + b ( x , t ) .
(4.5)
Since b(x, t) is small compared with B~(x) for a strongly-pinning sample so far as the usual observable time interval is concerned, we only have to retain the first-order terms with respect to b. Then eqs. (4.2) and (4.3) are reduced to
Ob
~{Bc(X)Uc(X)exo[ fob
OJc.~gUcdi(Uc)Vb(Uc)q)"
- ~x-/~°-~c°) '
0t
- ~ B c ) ~aL(Bc) Vc(X)--ar( [1--exp(FL(Bc)(tf(Bc)
4---~; T-
Vb(B¢))]
ks T
_]J''
(4.6)
(4.7)
where we have used Fp = J~ (B) B and FL = J ( B ) B = B (OB/Ox)/IZo. In the actual situation, some traces of local flux flow still remain, even when the variation of B due to flux creep becomes dominant. Then the definition of the time 6 at which the flux creep is observed as dominant has some ambiguity, as well as the value of the flux-creep velocity v~ at t = 6. It can be proved easily that eq. (4.6) has a solution of the separated type, b(x, t) = bx(x)bt(t), and that the resulting expression for the solution is given by
B(x, t) =Bc(x) + fl(x)ln( t/t~) ,
(4.8a)
where fl(x) is defined as Be
[fl(x)[ =kaTJ~(Bc)
J~(B-~c(B) '
Uc(B~) = 3jc(Bc)Bcd~(Bc) Vb(Bc) .
(4.8b)
(4.9)
K. Yamafuji et al. / Thermally activated flux creep
751
The actual value of Vc can be obtained from the relation d(Bcv~)/dx= -Ifl(x)l/tc. When the external magnetic field is fixed at Be, the magnetization M per unit volume is given by M = ( B ) x - Be where ( )x represents the average with respect to x. Then the flux-creep rate is given by BO
i=lOM/Oln(t/tc)l=l(fl(X))xl=p---o-o~~
Bc
dBc Be
f
Be
dBj~(B)U~(B), ,
where Bo is the value of the B~ at x = 0 . If the external field Be is much larger than Bp, with Bp denoting the characteristic field at which the flux front reaches the center of slab, i.e., Bp~-I~oJ~(Bp)Lx/2, then eq. (4.10) is reduced to a simpler form. For example, if we estimate Jc(Be) from the observed magnetization curve at Be>>Bp by ]Me(Be) ] --~toJc(Be)Lx/4, then the averaged flux-creep rate between the two processes, where the external field is fixed at B~ after increasing and decreasing the external field, is given by
R/IMc I =kB T/Uc(Be)
(4.11 )
•
Whereas this expression is formally the same as those in existing theories [ 8,10-13 ], the detailed expression for U~ is different from the previous ones. Furthermore, U~ is a given function of observable quantities in this theory. We like to emphasize that the simple relation given by eq. (4.11 ) can be obtained only for the above-mentioned process. Otherwise, the apparent effective pinning potential Uo(Be) estimated from R/IMcl = k~ T/Uo (Be) shows some deviation from Uc(Be). A detailed discussion on this point is given in appendix B.
5. Comparison of the present theory with observed data The expression for Uc(B), which is essentially related to the relative magnetic-flux creep rate as shown in eq. (4.1 1 ), is given by eq. (4.9), in which the average volume of each flux bundle is typically given by eq. (2.15). In actual observations, however, we sometimes encounter a situation, which deviates from the assumptions made in the derivation of eq. (2.15). These situations are summarized in table I, and the corresponding expressions for Vb are also shown. In table I, the cases (c) and (d) usually appear for a thin film when a magnetic field is applied perpendicularly to the film-surface. A detailed discussion for these cases will be presented in the next section. Whereas the modified Labusch parameter as (0) in eq. (2.14 ) approaches zero as J--'Jc [ 16 ], as (0) is nearly equal to the Labusch parameter aL defined by eq. (3.7) at J
(5.1a)
Uc (B) ~- ]rl[J~(B)B] ,/2 [di(B) ]3/2 [af(B) ]2[C44 (B) ]1/2 for case ( b ) .
(5. lb)
Table I Expressions for the average volume of a flux bundle, Vb, in various cases. Case
(a)
(b)
(c)
(d)
144
144
144>Lz
144>L~
2/66 >> ar
2/66 ~ af
2/66 >> af
2/66 < ar
~ (2/66)2/44
~a2144
~ (2166)2L~
~arZL~
situation
Vb
K. Yamafujiet al. / Thermally activatedflux creep
752
it is well-known empirically that the following relation holds for the interaction distance, di [8,23 ]:
d,(B) = 6 a t ( B ) ,
(5.2)
where the fluxoid spacing af(B) is given by
af(B)
{ 2 (/)o) '/2 -if/
= \~
(5.3)
with ~o denoting the fluxoid quantum. The coefficient 6 does not depend on the magnetic field B but depends on the pinning strength and on the superconducting parameters. Insertion of eq. (5.2) into eqs. (5. l a) and (5. lb) leads to the B-dependance of Uc(B):
U¢(B) ~_3rl65/2 [J¢(B)B] -,/2 [af(B) ]5/2C66(B) [C44(B) ],/2
for case ( a ) ,
(5.4a)
Uc (B) ~- ~t163/2[Jc(B)B ] x/2[af(B) ]7/2 [C44(B) ],/2
for case (b) ,
(5.4b)
where the expressions for C44(B) and C66 (B) are given by eqs. (2.17a) and (2.17b), respectively. Detailed measurements of flux creep in conventional superconductors suitable for a quantitative comparison with the present theory are rather scarce, because detailed information on the pinning characteristics necessary for the quantitative comparison with the present theory is not specified in most of the existing measurements. Antesberger and Ullmaier [25 ] investigated hollow cylinders of N b - T a alloys containing normal-conducting precipitates. For the sample in which B~2=0.6 T and x = 4 . 5 at 4.2 K, for example, Fp is 3.1 × l07 N / m 3 at Be = 0.24 T. The observation of Uc(B~) was carried out by averaging the creep rates between increasing- and decreasing-field processes after zero-field cooling. Since eq. (4.11 ) can be applied to this process because Be is much larger than Bp( ~ 0.05 T), the observed result yields Uc(0.24 T ) = 0 . 2 3 eV. Since the observation of d~ was not carried out by Antesberger and Ullmaier, we adopt the value of di=af/20, observed in a similar sample of N b - T a [26]. With this value ofdi, we find that case (a) is realized in the range 0.1 < B / B c2<0.9. Since Uc for case (a) is given by eq. (5.4a), we compare the theoretical curve with the observed data for Uc(B~), and the result is shown in fig. 5. The quantitative agreement is quite satisfactory. Another detailed measurement of flux creep in conventional superconductors is provided by Beasley et al. [ 12 ] for Pb-10% T1. For the second sample in which Be2 = 0.2 T and ~c= 2.7 at 4.2 K, for example, Jc is 3.0 × 107 A / m 2 at Be=0.1 T. The observation of U¢(Be) was carried out by averaging the creep rates between increasingand decreasing-field processes after zero-field cooling. Because of Be >> Bp ~ 0.02 T, eq. (4. l 1 ) is applicable and
0.5 0.~
2 rnple 1
o
o
Sample 2
:~ 0.2 o.1
0
0.2
0.4
0.6
0.8
B/Bc2 Fig. 5. Comparison of the theoretical curve for Uc(B¢)given by eq. (5.4a) and the observed data for Nb-Ta studied by Antesbergerand Ullmaier [26 ]. We estimate Ucfrom eq. (5.4a) using the observed value of JcB.
0.4
0.6
0.8
B/Bcz
Fig. 6. Comparison of the theoretical curve for U¢(B~)obtained by eq. (5.4a) and the observed data for Pb-10%Tl given by Beasley et al. [12]. The values of d~ are chosen as af/9.94, af/1 1.7 and af/17.0 for sample 1, 2 and 3, respectively.
753
K. Yamafuji et al. / Thermally activatedflux creep A
60
lOkG
50
3kG 2kG
E ~t-4 0
2s g
~20
~I " '\
o
IkG N
c
~
~;' II
{D
O
= -25
0
i
10
i
10 2 Time ( s e c )
i
10a
0 C
',,7
~' -so
J -10
-5
5
10
Magnetic Fiel0 (k6)
Fig. 7. (a) Creep rates of the magnetizationfor a quench and melt growthprocessedsample in remanent states providedby Murakami et al. [9 ], where an external field was increased up to Bm and then decreased to zero. The values of Bm are indicated in the figure. (b) Magnetization curves for the melt-processedsample of Lx= 1.l mm observedby Murakami et al. [9 ]. the observed result yields Uc(0.1 T)~- 1.0 eV and (3/4)Vbdi ~- 5.0× 10 -26 m 4. Since the observation ofdi was not carried out carried out by Beasley et al., we estimated the value of di from the observed value of (3/ 4) Vbdi ~-5.0 × I 0-26 m 4. If we assume that case (a) is realized, the best result is given by di ~ af/11.7. With this value of di, we get 2/66~ 0.92 ~tm while ar is 0.15 ~tm, which confirms the assumed situation of 2/66 >> af for case (a), as expected. The comparison of eq. (5.4a) with observed data of Uc(B) by Beasley et al. [ 12] are shown in fig. 6. The agreement is also quite satisfactory. The assumed values of di in fig. 6 are somewhat larger than the observed value ofd~ = af/30 for Pb-20% TI provided by Matsushita et al. [27 ]. We note that a broad peak effect in J¢(B) is observed in the range of 0.5<~B/B¢2~0.9 for Beasley's sample, while no peak effect is observed in Matsushita's sample. Since it has been shown [28] that the di takes a larger value when a peak effect appears, the above-assumed values of d~ seem to be reasonable. Flux-creep measurements on a strongly-pinning bulk samples of the high-T~ oxide superconductors have only been carried out by Murakami et al. on quench and melt growth processed slabs of Y - B a - C u - O at 77 K [9 ]. Their measurements were carried out at the full-remanent state obtained by fixing Be at Be = 0 after Be was increased up to Bm> 0.4 T ~ 2Bp, and the results are shown as the uppermost curves in fig. 7a, which yield values of U0 of about 0.6 eV. Although this observed process did not satisfy the required situation leading to eq. (4.11 ), the value of Uc estimated with the relations shown in appendix B, using the observed magnetization shown in fig. 7b, is not very far from Uo i.e., Uc(0.2 T) ~0.75 eV. This type of melt processed sample, however, has a special configuration in which several tens of grains are nearly separated by normal-conducting bands [ 9 ]. Thus the local critical current density Jog inside the grains probably plays a dominant role for flux creep, as in a sintered sample. The typical values Jog and d~ observed by us, using Campbell's method [8,23 ], are J~g~ 2 × 10 ~° A / m a and di ~ a f / 1 0 at 0.2 T for similar samples of this type of melt-processed Y - B a - C u - O . Is we adopt these values, the value of/66 ~ (C66di/JcgB)l/2 is much smaller than af at 0.2 T, and hence the present situation may be classified into case (b), 2/66~
6. D i s c u s s i o n
According to the present theory, the relative creep rate of the magnetization is typically given by eq. (4.11 ), though one should realize that Uo(Be) estimated from the observed value of_Rby the relation of/~/JMc] = k a T /
754
K. Yamafuji et al. / Thermally activatedflux creep
Uo(Be) is not always equal to Uc(Be), as shown in appendix B. Whereas eq. (4.11 ) is formally the same as the expressions given in previous works [ 8,10-13 ], the expression for Ue(B) given by eq. (4.9):
Uc(B) = 3jc(B)Bdi(B) Vb(B)
(6. la)
is different from the previous expressions [8,10-13]. Furthermore, the present expression for Uc(B) is a function of observable quantities. That is, Vb(B) is given by eq. (2.15) or table I, where the correlation lengths of pinned fluxoids are given by 144 -----(C44/Ol'L)1/2= (C44di/JcB) 1/2,
(6.1b)
166 ,.~ (C66/OLL) 1/2= (C66di/JcB) 1/2,
(6. lc)
with the C44 and C66 shown in eq. (2.17a) and (2.17b). c~L or d~ can be determined by Campbell's method [8,23] from the relation Fp=J~B=aLdi. As shown in the previous section, the present expression for Uc(B) can explain quantitatively the existing observed data [9,12,25] by assuming reasonable values for the interaction distance, d~(Be). No observation of d~(B), however, has been carried out for the sample in which the creep rate of magnetization was observed [9,12,25 ], and hence a more detailed check_,on the applicability of the present theory is desirable. It is to be noted that the expressions for Uem and Uem given by eqs. (3.8a) and (3.8b) are not applicable for J
Vcr=LvBeaf(Be)fo
{
1-exp
[ 4Clf(Be)Uc(Be)j
)
exp
Uc(Be) ( 1 Jc(Be)}_J
(6.2a)
where the thermal vibration frequency, fo (Be), of a flux bundle inside the effective pinning potential Ue is given by
aL Je(Be)pf(Be) f o - 2re(- 2nBedi(Be) "
(6.2b)
pf(Be) ~- (Be/Bc2)PN denotes the flux-glow resistivity, and we have used the relation (= B 2/pf(Be). The present expression for Vcr induced by the flux creep in thin film samples is somewhat different from the expressions presented in earlier works [3,8]. Furthermore, the present expressions forfo and Vcr are given functions of observable quantities including pinning parameters if we assume (tf(Be) ~-af(Be), as is usually done. It is to be noted [6] that, for estimating Uc(Be) from the observed data of Vcr using eq. (6.2a), all the quantities are functions of temperature except the external variables, Be and
K. Yamafuji et al. / Thermally activatedflux creep
755
The thermal vibration frequencyfo of flux bundles, given by eq. (6.2b), indicates that the motion of the flux bundles inside the effective pinning potential Ue, shown schematically in fig. 4, is a friction-damping motion triggered by thermal fluctuations. Since the value of 6 defined by di (Be) = ~ a f ( B e ) is typically of the order of ½n for high-x materials such as N b - T i [29] and melt-processed Y - B a - C u - O [9 ], 2nBedi(Be) ~ Bear(Be) is of the order of 10 - 7 m T at Be~ 5T. If we adopt the value o f p f ( 5 T ) ~ 1 0 - 6 ~ m for B i - S r - C a - C u - O [4] at 77 K, for example, )Cogiven by eq. (6.2b) leads to a value of the order of i 0Jc. Taking for Jc (5T) a value of 108~ 10'°A/ m 2 at 77 K, fo becomes of the order of 1 0 9 ~ l011 Hz, as suggested by Palstra et al. [4] and also by Malozemoff et al. [ 30 ]. A more detailed investigation is, however, necessary to check the applicability of the present theory. It is to be noted that the value of Uo ~ 0.6eV at 77 K for a quench and melt-growth processed bulk sample of Y - B a - C u - O [9] is not very small compared with the values of Uc observed in Pb-T1 and N b - T a at 4.2 K, as can be seen in figs. 5 and 6. Thus the main reason of the large flux-creep rates in high-To oxide superconductors observed at 77 K is the strong thermal activation at 77 K, as far as strongly-pinning bulk samples are concerned. While we have no observed data on the flux-creep rate in high-x conventional superconductors such as N b Ti at present, the value of C66 becomes smaller for strongly-pinning samples as can be seen from eq. (2.17b). Then case (b) in which 2/66 < af may be applicable, as in the melt-processed sample of Y - B a - C u - O [ 9 ]. In this case, eq. (5.4b) predicts that Uc increases with increasing Jc, although di=6af has some dependence on the pinning strength. Thus it is not unreasonable to expect a further increase of Uc for more strongly pinning samples of the high-To oxide superconductors. Glassy features due to random weakly coupled grains in high-temperature superconductors have been considered as another candidate for the mechanism of the magnetic relaxation [ 31 ]. However, agreement between the present theory and experimental data not only on new but also on conventional superconductors indicates that this magnetic relaxation can be systematically explained by the mechanism of thermally-activated flux creep. Recently, it was found [32] that the fluxoid lattice melts as liquid at high temperatures. The melting seems to appear at high temperatures where the irreversibility disappears [ 33 ]. Thus the region in which melting appears may be suppressed in a strongly-pinning sample. The tendency that the melting occurs more easily in B i - S r - C a - C u - O with smaller pinning forces than in Y - B a - C u - O seems to support this idea. However, the flux motion may be activated by the mechanical oscillations in the measurement of the fluxoid lattice melting [32]. This may lead to a difference from the usual flux creep due to thermal activation only.
Acknowledgment The authors would like to express their sincere appreciation to Dr. Masato Murakami and the research group in Nippon Steel Co. for showing their observed data before publication.
Appendix A The probability function P(u, t) for the relative displacement u obeying the Langevin equation (2.7b) is described by the following Fokker-Planck equation [22 ]:
~-i=~Jdr~u'L6u
kB
P,
(A.1)
where V is the volume of the semi-macroscopic region in which B can be regarded as nearly constant and ~b is the potential functional leading to the right-hand-side ofeq. (2.7b) by taking the variation of ~b with respect to u:
756
K. Yamafuji et al. I Thermally activatedflux creep
1 "+'C44
1
1
~
+
~
4
4+ 1
+(~y)
-[-C66\ ay 3t" a x J ) + ( C 1 1 - 2 C 6 6 )
1
~Y) .
(A.2)
Carrying out a statistical average of u using eq. (A. 1 ), we obtain eq. (2.10). Similarly, a statistical average of ux(r' +r, t)Ux(r', t) leads to the equation for the correlation function, S~x(r, t) [ 16]: O~
( 2
r
02
02
0 2 "]
Sxx(r, t ) = - a ~ ( t ) S , ~ ( r , t)+ICII~..2+C667.z+C44-~_21S~Ar, t) " k ox oy oz _1
+ (C~ -C66) ~
02
(A.3)
Sxv(r, t) + k B T r ( r ) .
We can get similar equations for S~y(r, t) and Sxy(r, t). The Fourier transform of this equation leads to the spectrum of the correlation function: k2 2 /~.,-.,'(k) ~'~"k B T~ 2~-
2
2
1
kx+k~ C,,(k~+k~)+C4~k~+o~(0)
kk ~
1
1
k~+k~ C66(k~+k~)+C44k~+OLs(0) '
(A.4)
Similar spectra for Z~y and X~ can be obtained [ 14,16 ]. Carrying out the inverse Fourier transform of Xx.~,we get eq. (2.13), where A~ (r), A2 (r) are given respectively by
1 (
(x2+y 2 Z 2"~ , ~ 2l 2, "~ A 1(r) --8~Cll/4 4 _ (1 +cos20) \ / - - - - ~ +/--~-4) *c°SZVx2-T~Y2,] ' Ae(r)=.8~6144
(X2+y 2 Z2"~ ( l - c o s 2 0 ) \ 126 +
..
(A.5a)
2/26
)-coszv ),
(A.Sb)
with 0=tan -l ( y / x ) .
Appendix B For a strongly-pinning bulk sample of a conventional superconductor, the magnetic-field dependence of the critical current density, Jc(B), can be approximated by [34 ] Jc(B) =Jc(Bp) ( B / B o ) r - ' [ 1 - (B/Bc2) 1'/[ 1 - (Bo/Bce) ] ' ,
(B. la)
where ? and e are numerical pinning parameters taking a value in the ranges of 0 < 7-< 1 and 1 -< E< 2, respectively, and Bp is the characteristic field at which the flux reaches the center of the sample. If we confine the present discussion to the range of B of B¢~ << Bp < B << Be> for simplicity, eq. (B. 1a) is reduced to J A B ) = J A & ) (B/Bp) > ' •
(B.lb)
For the case of a slab considered in section 4, therefore, eq. (4.4) is reduced to 052-:702= v,
(B.2)
where v is the sign factor taking the value of +_1 according to the direction of the Lorentz force, and 2 and /~ are normalized quantities: 2=2x/L,-,
~=B/Bp;
Bp=#oJc(Bp)Lx/2.
(B.3)
K. Yamafujiet al. / Thermallyactivatedflux creep
757
With the aid of these normalized quantities, eq. (4.10) is reduced to
ka T
bo
bc
Ue(Bp )
R--Uc~p)BP f d b c f d~l-TVc(bap ) ' he be
(B.4)
where we have used eqs. (B.lb) and (B.3). Similarly, eqs. (5.3a) and (5.3b) lead to
Uc(bBp) B ~ ( 3 _ 2 7 ) / 4 Ue(Bp )
for case (a) ,
(B.5a)
Ue(bBo) -~ 5(27_3)/4
for case (b) .
(B.5b)
Uc(Bp)
Then eq. (B.4) yields
R-
kBT B 4(2-7)(5-2Yb(e9-27)/4-b(~S-2Y)/4bo+ Ue(Bo) P 5 - 2 y \ 9 - 2 7
4---~b~9-2y)/4~ 9-27
u
]
R - kBT B 4(2-7)13+27b~7+z~,/4_b~3+2~)/4bo+ a .__£~b(7+2,~,/4 S -U~(Bp) p 3 + 2 y \ 7 + 2 y 7+2y e ]
forcase (a),
forcase (b)
(B.6a)
(B.6b)
•
On the other hand, the magnetization in the critical state is given by
Mc= ( Bc(x) ).~-Be =BpI 23~ ( b3-y-~3o-Y)-be l ,
(B.7)
where/~o is defined as
b?~-~'=b~-~- v.
(B.8)
Then the creep rate of the magnetization is given by
R/IMcI=kBT/Uo(Be); where
g(be)
Uo(Be)=Uc(Bp)/g(be),
(B.9)
is defined as
g ( b e ) = - - -1- 4(2--y)(5--2Yb(9_2y)/4_h(5_2y)/4/h2_y [m[ 5 - 2 7 \ 9 - 2 7 e ~e ,~e
l))l/(2_y)..[_ 4__.~(b2_7,_v)(9_2y)/4(2_~,)"~ --
9-2y
]
for case (a) ,
g ( b e ) - -Iml -1- -
4(2-y).(3+2Yb(7+2~)/4_].l(3+2y)/4(h2_~ 3+29, \ 7 + 2 y ~e ,~e --
(B.10a)
v)l/~2_v>+7+_~7)(b~_y -
p)(7+2?)/4(2_~,))
for case (b) ,
(B.10b)
with the definition of m:
m=23~[b 3-y- (b(e2-y- v)(3-y)/(2-y) ] - b e . For example, when Be is stopped at Bp in an increasing process of v = + l,
(B.11)
g(be)
is given by
758
K. Yamafuji et al. / Thermally activated flux creep
g~(1)-
4(2-7)(3-7) 9-27
for case (a) ,
(B.12a)
gi(1)-
4(2-7)(3-7) 7+27)
for case (b) .
(B.12b)
On the other hand, when Be is stopped at zero in a decreasing process o f v = - 1 (i.e., for the remanent state), we obtain 16(3-7) go(0) = ( 5 _ 2 7 ) ( 9 _ 2 7 ) g0(0)=
16(3-7) (5-27)(9-27)
for case (a)
(B.13a)
for case ( b ) .
(B.13b)
For practical conventional superconductors such as N b - T i and Nb3Sn, it has been known that the value o f 7 is nearly equal to ½ [ 35 ]. For this case of 7 = ½, both eqs. (B. 12a) and (B. 12b) lead to g~( 1 ) = ~-~ so that we have Uc(Bp)=~Uo(Bp). Both eqs. (B.13a) and (B.13b) lead to g o ( 0 ) = ~ for 7=½, so that we get Uc(0) = ~ Uo(0) for the full remanent state. Since the magnetization curves shown in fig. 7b are quite similar to those for 7= ½, we can use this ratio of Uc(O)/Uo(O)=~. It is to be noted that the relation of Uc(Bp) to Uo(B¢)becomes more complicated in the range of B<~Bo2, though we are able to derive the required relation with the present scheme.
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