Supercond. Sci. Technol. 1 (1989) 242-248. Printed in the UK
P H Kes, J Aarts, J van den Berg,C J van der Beek andJ A Mydosh
Kamerlingh Onnes Laboratorium, Leiden University, PO Box 9506. 2300 R A Leiden,
The Netherlands
Received 26 September 1988
Abstract. The theory for thermally assisted flux flow (TAFF) in the limit of small
driving forces is used to derive exact expressions for the time-dependent behaviour
of the magnetisation and permeability. The problem is especially relevant for the
high-temperature superconductors where large variationsof the transition
temperature in small DC fields are observed as a function of the frequency of the
probing AC field. The parameters of the theory are extensively discussed in relation
to the present understanding of flux pinning.
1. Introduction
It has been observed recently that the phase transition
temperature of the superconducting oxide YBa,Cu,O,
in an applied DC field depends strongly on the frequency
v of a small superimposed AC field [l, 21. Malozemoff
and co-workers have argued that the real critical temperature in a field T,,(B) is not observed, but rather that
experiments determine a lower temperature denoted as
an irreversibility temperature Trr [l]. At Trr the permeability shows a maximum in the out-of-phase signal
p" and a steepest decay in the in-phase component p'.
The maximum will occur in thetransition
regime
between almost reversible fluxflow
above Trr and
strong irreversible screening due to flux pinning below
T,, . In the range Trr< T <
flux lines are depinned
by thermalactivation,a
process we call thermally
assisted fluxflow (TAFF) to discriminate it from flux
creep. Although both processes have thesame origin
and are in principle described by thesamecontinuity
equation,thisequationcan
be linearised and solved
analytically only in these two limiting cases.
Flux creep is the phenomenon that occurs when the
driving force is almostequal to the pinning force, i.e.
just below the onset of the usual flux flow regime. TAFF
only occurs in superconductors with an intrinsically low
pinningbarrierand
at temperatures high enough to
overcome this barrier, and is therefore observed already
in the limit of small driving forces. The importance of
TAFF for the high-temperature superconductors was first
pointed out by Dew-Hughes [3]. Herethe
ratio of
pinning potential height' U to thermal activation energy
kT is much smaller than in conventional superconductors, U/kT < 10 rather than 100. In the accompanying
.experimental paper [4] it is shown that such effects are
clearly manifested in the Bi compound with a T, of
87 K. The shift of the transition temperature with fre-
T,,
0953-2048/89/050242
+ 07 $02.50 @ 1989 IOP Publishing Ltd
quency is much larger thanYBa,Cu,O,,
6 K per
decade instead of 0.7 K per decade [17]. This large difference may be related to the effect of twin planes [S].
Related to this frequency dependence is the unusual
low value of the field B,(T), at which the irreversibility
in themagnetisationdisappears, especially when measured with the field along the c axis of the crystal [6].
The relation B,(T) defines the 'irreversibility line', which
has been formerly described as Trr(B)[l].
In previous work [l, 2, 71 attempts have been made
to model Trr(B) and Trr(v) by intuitively constructed
expressions. In this paper an exact derivation is presented from which these effects can be obtained in principle.
It is argued that the results depend sensitively on the
pin mechanism and the choice of parameters. Therefore
fitting results should be considered with some care, as
long as detailedinformation about the pinning-force
density is lacking, partially because TAFF does not allow
a straightforward determination of critical current and
flux pinning effects [6]. On the other hand, references
[l, 2, 71 serve to point out the importance of TAFF and
the consequences it may have on the interpretation of
importantparameters for thesuperconducting oxides,
like the upper critical field B,,.
The outline of our paper is as follows. After
developing the theory in $2, several possible choices for
the parameters will be discussed in $ 3 based upon our
present knowledge of flux pinning.
2. Theory
For small values of the parameter U/kT the flux creep
theory [S, 91 in the presence of a driving force must be
extended to consider the possibility of reverse hopping
[lo]. Because of flux conservation the flux density B is
Thermally assisted flux flow
described by the continuity equation
at = V
[
( z ) B w v , exp( )2-;
sinh(
g)],
(1)
Here the notation of [S] has been adopted in which
v, is the attempt frequency, W the distance by which a
certain volume V , of the flux line lattice (FLL) moves in a
thermally activatedjump, U the energy required to
depin the volume V,, and A W the work done to move
this same volume against the driving force. In $ 3 relevant choices for V , and W willbe discussed. It willbe
shown there that U can be related to the critical current
density J, , and A W to the actual currentdensity J by
U = JcBV,rp
J
J
c/ \rp/
Both U and AW are functions of B and T, and rp is
the range of the pinning potential. For defects acting as
pinning centres, with the relevant dimensions smaller
than
the
Ginzburg-Landau
coherence length, ((T),
rp x for isolated vortex cores, i.e., b = B/BC2c 0.2.
Alternatively, rp z a,/2 for overlapping vortex cores, i.e.
b > 0.2 [1l]. Otherwise the defect size would be a more
appropriate measureof rp.
Beasley et a1 [S] treatedthe flux creep sitation in
which U is very much larger than kT, so that the effects
are only observable for (J, - J)/J, < 1. In this case
it follows that 2sinh(AW/kT) = exp(AW/kT). We,
however, will solve equation (1) in thesmall-J limit
(TAFF), where 2 sinh(AW/kT) = 2AW/kT. This factor in
equation (1) generates the driving force which is simply
J B and J = p G I V x B, because for the hightemperature superconductors ic (the Ginzburg-Landau
parameter) is large and H,,small [l2], so that p * , the
slope of the reversible induction curve, may be set equal
to unity.
Two geometries areconsidered:aslab
defined by
-1 < x < l and a solid cylinder r < a, both with their
surfaces parallel to the field direction which is along the
z axis. After substitution equation (1) becomes
H(t) = H,
(ii)
H(t) = H , 2 - f i t
1
a
(7)
(8)
which describe, respectively, the AC experiments and the
DC magnetisation measurements (irreversibility line).
Thesolution is outlined for theslabgeometry;
full
details are given in appendix 1. In case (i) the solution
is SB(x, t ) = A(x)sin(2nvt &)). Theamplitude A(x)
decays approximately exponentially with a decay length
+
k”
(9)
= (Do/.~)1’2.
1 sinh 2u
PI = 2u cosh 2u
cylinder
+ sin 2u
+ cos 2u
(10)
1 sinh 2u - sin 2u
=-
<
aB
+ h sin 2nvt
(i)
The average induction follows from integration of x
over the slab and yields the expressions for both permeability components:
w
A W = U( T\)T
(
)
.
\
tions of equations (4) and ( 5 ) for many geometries and
boundary conditions can be foundin [ 131.
Two applied field modes have been considered here:
2u cosh 2u
+ cos 2u
*
The results depend on temperature via the variable
= ( ~ v I ~ / D , ) ”The
~ . functions $(U) and pff(u),as
obtained in appendix 1, areshown in figure 1. The
maximum in p f f for the slab geometry is attainedat
M, = 1.1275. By substitutingtheaboveintoequation
(6) an expression is derived that relates the temperature
and field at the maximum of p’’ to the frequency v of the
AC field :
U = kl
U = kT In
2Miv, BUw2
nvp, J, kTl2rP
Analogously, for the solid cylinder we obtain
MEv, BUw2
nvp, J, kTa2rp
U = kT In
It is expected that M, will also be of order unity. In
addition, the definition of the decay length in the case of
(5)
with
D =~
(”) (t)
W V k
, T
( ~ )
Po J,
ex,( -
3).
(6)
D is related to the flux-flow density D defined in [S]
and is analogous to the diffusion constant in a diffusion
problem. It is obvious that D depends on B and T and
is further influenced by the pin characteristics. However,
since J is taken to be small over the entire superconductor, B is almost uniform, i.e. B = B, + SB with SB 6
B , . It is shown in appendix 3 that, for SBO.lB,,
D
may be replaced by its constant value D , D(B,). Solu-
1
0
I
I
I
1
I
1
2
3
4
5
U
Figure 1. Graph of the permeability components against the
parameter U .
243
P H Kes et a/
a cylindrical geometry differs by afactor ,b, which
explains the missing factor of two in the argument of
the logarithm.
These results differ from the intuitive equation suggested in [7]. For instance, because of the exponential
decay, our present results do not depend on the amplitude of the AC field. This is in agreement with our
experiments on the ‘B? single crystals [4]. In fact, the
difference in the arguments of the logarithms is a factor
B,w/GBa. Fortuitously,thisfactor
is of order unity,
although both 6B/Bo and w/a may be of order
The physics of the problem is clearly demonstrated by
the plots of A(x) and C#J(X)for various kl given in [13]. If
kl = 0.5 corresponding to a relatively large D,, the
behaviour is almost reversible. If kl = 10, the screening
already amounts to 90%. It also follows that M , and
M , must be of order unity.
The solution for case (ii) is relatively easy (see [13])
and gives the magnetisation M = @po - H as
for the slab. Here
7, = z0/(2n
+ l),
T, = 412/712D0.
(15)
(1 6)
Assuming that the
measurement
is carried out
slowly enough to neglect the transient part in equation
(14) (see appendix 2 in which the response to a step-like
change in field is treated), a condition is derived for the
field-temperature relation at which, in decreasing fields,
amagnetisation signal can be observed for the first
time. For high-rc materials the reversible magnetisation
below H , , is negligibly small, M x ( H , , - H)/2rc2.
Therefore, the condition can be expressed in the form of
a ‘minimal-observable’ current density Jmin= 2 I M [/l,
which dependsontheexperimentalresolution.The
‘irreversibility line’ should thus follow from
U = kT In
for theslab.
which yields
3Vo B U W ~ J , , , ~ ,
p, HkTlr, J ,
For the solid cylinder Jmin= 3 I M [/a,
U
=
kT In
16v0 BUw2Jmi,
3p0 HkTar, J ,
If the measurements are carried out in pulsed magnetic fields [S], it should be carefully checked whether
the above analysis is still applicable. For high field rates
it is more likely that a current density closer to J , is
probed. In that case analytic solutions of equation (1)
may not be available, so that numerical methods should
be used.
Equations (12), (1 3), (17) and (1 8 ) serve as a starting
point for the interpretation of the TAFF effects. It is clear,
however, that even a qualitative description depends on
the waywe choose the parameters and
this requires a
detailed understanding of the flux-pinning mechanism
as will be discussed in the following section.
244
Finally, it should be noted that the above theory
describes TAFF when the driving force is caused by a flux
density gradient. The situation of TAFF due to a transport current shows a striking formal analogy with the
preceding result, since from the formalism of [3] it
follows that the flux flow resistivity pf can be expressed
as
Pr = PODO
with thesame D, as in equation (6). Bothkinds of
experiment, therefore, provide comparable information.
3. Discussion
For comparison
a
with experiment, the physical
meaning of theparametersin
the theoryshould be
understood. In general, the parameters V,, r , , W , v, and
F , = J , B can be well defined. In practice, however, it is
the defect morphology of the superconductor that determines thecharacter of thepinningcentres
and the
actual assignment of theparameters.Thelatter
is a
complicated problem and leaves many possibilities for
the ultimatetemperatureand
field dependence, especially when it is not obvious what the predominant pins
of thematerialare.Inthe
following discussion we
restrict ourselves to pins that have at least one dimension, namely that in the direction of the driving force,
L of the pins
smaller than r p . Themutualdistances
should form a random distribution.Theirinteraction,
the so-called coreinteraction, is through local perturbations of the mean free path or the coupling constant,
which couple to the modulation of the superconducting
order parameter.
3.1. The volume V,
The theory of flux creep actually deals with the problem
of hopping over a potential barrier by independent particles driven by an external force and thermally activated by theinteraction
with aheatbath,as
in the
treatment of Brownian motion by Kramers [14]. As has
been already recognised by Anderson [S], the analogue
of an independent particle in a FLL is a ‘flux bundle’
with volume V , that is elastically independent of the
other ‘particles’ of the FLL. This volume is the same as
the correlated region in the theory of collective pinning
[15]. It is the volume in which the FLL maintains its
positional short-range order, which is either determined
by the correlation lengths R , and L,, or by the density
of dislocations in the FLL, or more generally, the
amount of disorder of the FLL [16]. In this sense it
might be that the interaction with the thermal bath contributes to the disorder and the size of V , [17].
The choice of V , for an actualpinningproblem is
not unique. In the case of a thin superconducting film
with many small pins it can be predicted [18-201. For a
system of grain boundaries which exert a very strong
elementary interaction on the FLL one can make a reasonable
estimate
[21, 221. The final result for V,
Thermally assisted flux flow
depends on the strength of the elementary interactionf,
or the elementary pin potential R,, and the distance L
between the pins. Since many details are not yet known
for the high-T, superconductors, it is impossible to
make a unique choice, so that several possibilities can
be considered. One should realise,however, that the
study of collective pinning on thin amorphous films has
revealed that the FLL usually is highly disordered even
for very weak pins [20]. It is therefore assumed that the
‘amorphous’ limit is appropriate for most strong
pinning systems. In this limit V , x a; L , , where a, =
1 . 0 7 5 m B is the FLL parameter and L, is the correlation length along the flux line. As argued in C231 and
[24] it is not clear how L, can be predicted for point
pins from theory. Several approaches can be used. In
[l51 a formula was derived which in the clean limit
(to> I t r , with to the BCS coherence length and I,, the
transport mean free path) and for b > 0.2 can be cast
into the form
where 1 is the penetration depth and W
below). For instance (with b > 0.2),
= n( f 2,
(see
for dislocation loops, and
W, = C,BC2e b 3 ( l - b)2
(21)
forvoids, and probably also for precipitates [18, 241.
Using a somewhat different elastic constant for tilt,
which turned out to be veryuseful in predicting the
transition from two- to three-dimensional disorder [24],
one arrives at
L, x. (Q: b2(1 - b)’)”)
4112p; A4 W
3.2. Relation between F,, rpand U
Once V , has been established one can obtain the
pinning force density F , by summing the individual
contributions of all the pins in V,. For a dense system of
randomly distributed pins this leads to a net force on V ,
which is equal to the fluctuation in the total effect of the
pins in the correlated region ( t ~ V , ( f ~ ) )where
’ / ~ , n is the
concentration of pins and the average should be taken
over the entire correlated volume. F , then follows by
dividing through V , .
If V , contains only one very strong pin a direct summation is more appropriate.One
can imagine,for
instance, that the strong interaction of the FLL with the
boundary of a grain will probably confine V , to the
volume of the grain itself, providing it is not too large
or contains other strong pins.
The pin potential in which this ‘particle’ istrapped is
now @veri by U = FpV,I , where, as in equation (2), I ,
is the range of the potential. In $2 we stated that rP is
either 5: or a,/2, but such is only true when pure collective pinning is encountered [18].If many flux-line dislocations determine the disorder, r, is a smaller fraction
of 5 (or a,) corresponding to the distance overwhich
the volume V , can be reversibly moved in its potential
[25, 261.
3.3. Attempt frequency and hopping range
Due to thermal motion of the pins the values of F,, and
probably also V,, fluctuate in time, causing the ‘particle’
to hop over the barrier. It seems reasonable to relate the
attempt frequency to the relevant dimensions of V,.
Using a sound velocity of 3 x lo3 m s - l , the typical
valuefor one fluxline to hop would range between
v , = 10” and3 x 10” Hz for corresponding fields
between 20 mT and 2 T.
As has been discussed by Kramers [l41 the viscosity
of the system damps the motion of the particle. This can
beexpressed as a reduction in the attempt frequency.
For the flux-creep situation the viscous damping
becomesincreasingly important because the effective
barrier heightdecreaseswith
larger driving forces. In
the flux-flow state ( J > J,) the viscosity determines, and
is inversely proportional to, the flux flow resistivity.
The hopping range W for a dense system of pins can
be taken equal to I , . This is because moving the lattice
over a distance of the order r, destroys the original
short-range order in the correlated region, which has to
be redefined. The same holds forall kinds of pin distributions when b > 0.2, because the potential then has
the sinusoidal variation of the superconducting order
parameter. Only for b < 0.2may considerable deviations between W and r, occur. At large a, the order
parameter changes only at the location of thevortex
core in a range 5: < a,. If in this situation L > a, and
the interaction is strong, so that there is only one large
pin per V,, then the hopping distance willbeof order
a,, while r, x 5:. And W x L if 5 < L c a, under the
same conditions. The discrimination between W , I , in
terms of a,, t and L isof importance if one wants to
compare experimental results with the TAFF expressions.
Experimentally, the temperature and field dependence is
investigated intheregionclose
to T,, so that every
factor t contributes a factor ( 1 - t)”” to the expression, whereas a, contributes a factor
3.4. Application to YBa,Cu,O, and Bi,Sr,CaCu,O,
Wenow apply the above ideas toa single crystal of
YBa2Cu30, with the field parallel to the c axis. In this
situation we presume that the twin planes dominate the
pinning but there is a significant background pinning as
well. This follows from the large critical currents measured in the other orientation, i.e. with the field in the
ab plane. The background will determine the length and
width of the correlated region, but the distance between
the twin planes forms the other dimension. Namely, if a
flux line, which is pinned at the twin plane, jumps loose
the whole row behind it moves to fill the vacancy. So
245
P H Kes et a /
V , = a, L, L and, because the experiments under
discussion arecarriedoutat
lowfields (but still with
L ( % 1 0 0 nm) > a,) I , = ( and W = a,. Therefore
F, = f p J L a , [S] with f,, x 0.5p0H:(0/ptr; thereby we
obtain U = fpl L, ( a (1 using equation (19)
for L, and [(l - t)5/B2]116when equation (22) is substituted. The argument of the logarithm in equations (12),
(13), (17) and (18) is proportional to (1 - t)li2/kTB in
the former case above and to B’I6/kT(1 - t),I3 in the
latter.The experiments show the (1 - t)3i2/B scaling
c11.
As a second example we apply our model to
Bi,Sr,CaCu,O, single crystals. Now there are no twin
planes that can contribute to the pinning force. In [4] it
is shown that pinning is much weaker andthe TAFF
effects are considerably stronger. Consider pinning due
to
point
defects in the isolated vortex limit. W,
(equation (21)) should then read [1 l]
W, = C:’B:B
(-
Combining these expressions with the collective pinning
a (1 - t). So U only
theory gives U = W,’12V:i2rp
depends on temperature and not on field, which seems
natural for isolated vortices. This temperature dependence is indeed observed [4], but the exact scaling with
field is not yet known. Such behaviour has to be studied
in future work. Our purpose in this last section was to
put forward the relevant considerations and procedures
necessary toarriveatthe
expression for U andthe
argument of the logarithmic term.
4. Summary
The flux-creep theory has been investigated in the limit
of low driving force. Expressions were derived for the
permeability in a magnetic field near T, and the magnetisation close to B,, at lower temperatures when these
quantitiesarepredominantly determined by thermally
assisted fluxflow (TAFF). The response of the magnetisation to a field step has been treated separately
(appendix 2). This theory provides the exact formulae
with which to analyse the experiments. The physical
interpretation of the parameters occurring in the theory
has been discussed and special attention paid tothe
relation between the thermally assisted effects and the
known principles of flux pinning theory.
Acknowledgments
P H Kes wishes to thank A P Malozemoff for many
helpful discussions and H CFreyhardtand
his colleagues at the Institut fur Metallphysik in Gottingen for
their hospitality duringthe time this paper was pre246
Appendix 1. Derivation of the formulae for U
The derivation of equations (12), (13), (17) and (18) is
given explicitly for the slab geometry - 1 < x G 1. The
solution for the infinitely long solid cylinder involves
much morealgebra and is therefore postponed for a
future publication.
(i) With H(t) = H, h sin 2nvt the solution for the
induction increment according to [13, p 1051 is
+
&?(x, t ) = A(x) sin(2nvt
(23)
and L, (equation (19)) becomes
L, x
pared. This visit was made possible by a research grant
from DFG Sonderforschungsbereich 126, project E10.
We also acknowledge financial support from the Dutch
Foundation for Fundamental Research on
Matter
(FOM).
+ &))
(A. 1)
in which /l= 6B/po h , and A(x) and 4(x) are the amplitude and phase of the function
Z(x) =
cosh kx(1 + i )
cosh kl(1 + i )
(A4
Theinduction increment averaged over x determines
the permeability components via
p’ =
f [Re(Z)
dx
(A.3)
p” =
f [Im(Z)
dx
(A.4)
from which equations (10) and (1 1) follow in a straightforward way.
(ii) For H(t) = H,, - kt the solution of B(x, t ) for
bothslaband
cylinder can be found in c133 PP 104,
2013 :
B(x, t ) = B, -
-
p,
k(12 - x,)
200
16p0
HI2
n3D,
.f
n=O
cos (2n + 1)nx
((-(2n exp(t/r,)
+
21
1)”
(A.5)
for the slab - 1 < x < l with B, = p, H(t) and 7, defined
in equations (15) and (16). Computing the average and
subtracting H yields equation (14) for M(t). Thesituation in increasing field is less well defined because we
are returning from the critical state. But at the criterion
for the irreversibility line theinductiongradient is so
small that we may safely assume that I M ( t ) I is the same
as in decreasing field.
Theanalogousderivation
for the solid cylinder
r < a yields the expression
B(r, t ) = B , -
p.
h ( a z - rZ)
40,
Thermally assisted flux flow
and
dicts a slope independent of temperature, which would
be observed aroundacharacteristic
time 2,/2, and
this time would vary strongly with temperature via
D,(T). The results of [2] are probably better described
by the flux-creep approximation of sinh(AW/kT)
exp(AW/kT); although closer to T,, the behaviour given
by TAFF should be observed.
Recently, evidence has been obtained [29] that the
magnetisation in single crystals decays according to a
stretched exponential:6M/6H0 = exp[-(t/~)~]. It can
be shownthatequation
(A.8)isnicely
described in
terms of a stretched exponential with B x 0.6 and T x
zo . In carrying out such experiments one should beware
of the fact that the initial flux distribution can influence
the precise time dependence observed afterwards.
-
J o and J , are Bessel functions and the a,, are the positive roots of J,(aa) = 0.
Appendix 2. Response to a field step
Solutions similar to the transient term in equations (14)
and (A.7) occur in the response of M after a step-like
increase of the field by an increment 6H.We assume a
uniform induction at t = 0 and obtain for a slab (see
c139 P 1@u
Appendix 3. Restrictions for D, and J
+
with the same definitions for z, = z0/(2n l)' and zo =
4 l 2 / n ~ ,as given previously.
In figure 2 6M in units 6 H o is plotted as a function
of t expressed in units of 7,. The inset shows a part of
the same graphonan
expanded scale. The decay
already sets in roughly three decades below z, due to
the higher z,, terms. As is seen from the above formula,
however, thecontribution
of these terms decreases
n.
At
about 22, the relaxation
rapidly with increasing
has reduced Mto10% of the initial value. The inset
clearly shows that there is no time interval in which 6M
changes linearly with ln(t). Such a time dependence is
observed in [2] for YBa2Cu,0, single crystals, but
otherbehaviourhas
been reported as well for this
material [27,281. Yeshurun and Malozemoff [2] also
reported a reduction of dM/d ln(t) for T approaching
T,. Such a behaviour cannot be described by equation
(A.2), as follows from the scaling. This expression pre-
1.0
-
0.8
-
0.6
-
The condition for approximating D by D, is worked out
for theslab
geometry, but is not sensitive to the
geometry. D is defined in equation (6). The B-dependent
part can be written as w2V, B2 exp( - U / k T ) . As has
been discussed in $3, it is not possible to give a general
relation for the B dependence. Yet, the most likely possibilities for thehigh-temperaturesuperconductors
at
relatively small fields (b < 0.5) are W a, or (, V , U:
or ai and U U. + cBp with U , the pin potential at
B = 0, some coefficient c and small power p 3 or 1.
The factor w2V,B2is thusproportionalto
Bs with
-
-3
-
-
c S c 1.
Substitution ofBs exp[ - U(B)/kT] into equation (4:
leads to the conclusion that
(x)
dB ' ~ B ~ ( s - PU+- U
)
-l
(A.9
E
P
2 0.4-
0.2-
0I
10-
I
lo-&
I
I
I
100
102
I
I
IO4
1o6
f IT0
Figure 2. Time-dependent decay of the magnetisation after a field step in the
limit of small driving current plotted on a normal scale (full curve) and a largely
expanded scale (broken curve). The inset shows the log T derivative.
247
P H Ker, el el
must be obeyed. A reasonable condition to allow for the
J x 0 limit is (U - U,)/kT
10, so equation (A.9)
becomes
(A. 10)
The solutions for B(x) studied in case (i) are of the kind
B = B, + dB,(cosh(kx)/cosh(kl)), hence we arrive at
dBo 4 O.lB,. F o r case (ii) thecondition is similar:
dB, 4 0.05Bo.
In addition, the maximum of J, J(x = I), should be
small enough for AW/kT -4 1. This requirement is fulfdled if J -4 (kT/U)J, x O.lJ,, which can be recast in
the form kl = U 4 kTJ,l/Uh = kTAB/UGBo, where AB
is the maximum flux density difference that the critical
state would create. This should be checked retrospectively. From the analysis for the Bi compound, con[4], the
above
tained in the
accompanying
paper
requirement is easily fulfilled.
References
[l] Malozemoff A P, Worthington T K, Yeshurun Y,
Holtzberg F and Kes P H 1988 Phys. Reo. B to be
published
[2] Yeshurun Y and Malozemoff A P 1988 Phys. Rev. Lett.
60 2202
[3] Dew-Hughes D 1988 Cryogenics to be published
[4] van den Berg J, van der Beek C J, Kes P H, Mydosh
J A, Menken M J V and Menovsky A A 1989
Supercond. Sci. Technol. 1 249
[S] Kes P H 1988 Physica C 153-155 1121
[6] Kes P H, van den Berg J, van der Beek C J, Mydosh
J A, Roeland L W, Menovsky A A, Kadowaki K and
248
de Boer F R 1988 Proc. Latin-American Con$ on
High Temperature Superconductivity, Rio de Janeiro
[7] Malozemoff A P, Yeshurun Y, Yandrofski R M,
Holtzberg F H and Dinger T R 1988 Preprint
[S] Anderson P W 1962 Phys. Rev. Lett. 9 309
[S] Beasley M R, Labusch R and Webb W W 1969 Phys.
Reo. 181 682
[lo] Tinkham M 1975 Introduction to Superconductioity (New
York: McGraw-Hill)
[l11 Brandt E H 1986 Phys. Rev. Lett. 57 1347
[l23 Umezawa A, Crabtree G W, Liu J Z, Moran T J, Malik
S K, Nunez L H, Kwok W L and Sowers C H 1988
Preprint
[l31 Carslaw H S and Jaeger J C 1959 Conduction of Heat in
Solids (Oxford: Clarendon)
C141 Kramers H A 1940 Physica 7 284
[l51 Larkin A I and Ovchinnikov Yu N 1979 J . Low Temp.
Phys. 34 409
[l61 Kes P H 1987 IEEE Trans. Magn. MAE23 1160
[l71 Nelson D R 1988 Phys. Reo. Lett. 60 1973
[l81 Kes P H and Tsuei C C 1981 Phys. Rev. Lett. 47 1930;
1983 Phys. Reo. B 28 5126
Wordenweber R, Kes P H and Tsuei C C 1986 Phys.
Reo. B 33 3172
[l91 Pruymboom A, Berghuis P, Kes P H and de Schepper L
1987 IEEE Trans. Magn. MAG23 942
C203 Wordenweber R and Kes P H 1986 Phys. Reo. B 34 494
C213 Pruymboom A 1988 Thesis Leiden University
C221 Kes P H, Pruymboom A, van den Berg J, van der Beek
C J and Mydosh J A 1988 in preparation
C231 Kes P H and Wordenweber R 1987 J . Low Temp. Phys.
67 1
C241 Wordenweber R 1987 Thesis Leiden University
C251 Campbell A M 1978 Phil. Mag. B 37 149,168
C261 van der Meij G P 1984 Thesis Leiden University ch. 6
[27] Hagen C W, Bom M R, Griessen R, Dam B and Veringa
H 1988 Physica C 153-155 322
[28] Yeshurun Y 1988 Private communication
[29] Hagen C W, Salomons E, Griessen R and Dam G 1988
Preprint