INSTITUTE OF PHYSICS PUBLISHING
SUPERCONDUCTOR SCIENCE AND TECHNOLOGY
Supercond. Sci. Technol. 14 (2001) 346–352
www.iop.org/Journals/su
PII: S0953-2048(01)18521-0
Effect of flux creep on Ic measurement of
electric transport
Y H Zhang1,2 , H Luo1 , X F Wu1 and S Y Ding1,3
1
National Laboratory of Solid State Microstructures, Department of Physics,
Nanjing University, Nanjing 210093, People’s Republic of China
2
Department of Materials Science and Engineering, Tongji University, Shanghai 200092,
People’s Republic of China
E-mail: syding@netra.nju.edu.cn
Received 2 November 2000, in final form 13 March 2001
Abstract
The influence of flux creep on Ic measurement was studied by numerically
solving the nonlinear flux creep equation at different dI /dt, Vc and n∗ ,
where dI /dt is the sweeping rate of the applied current (I ), Ic , Vc and n∗ are
the critical current, the criterion and the material parameter, respectively. It
is shown that the V –I curve consists of two parts converging at Ip , Vp and
(dI /dt)p at which the current fully penetrates the sample. In the segment
where I < Ip , the V –I curve is parabolic, V ∝ I 2 , and independent of n∗ ,
∗
whereas in the segment where I > Ip , the curve is a power law, V ∝ I n ,
reflecting the material equation. It is suggested that the appropriate region in
the V –I curve to determine n∗ is where I > Ip . Based on the V –I curve, it
is concluded that if dI /dt > (dI /dt)p , Ic decreases with increasing dI /dt.
On the other hand, if dI /dt < (dI /dt)p , Ic is independent of dI /dt and is
therefore suitable. The three critical parameters ((dI /dt)p , Vp and Ip ) are
dependent on each other. The parabolic V –I relation can be observed in the
giant flux creep state (small n∗ ), whereas in the critical state (large n∗ ), V in
the parabolic V –I relation is too small to be detected by a standard
voltmeter. This also indicates that the pulsed method may underestimate Ic
in the case of high temperatures or in strong applied fields, but it will not
affect Ic in the case of low temperatures, weak applied fields and strong
pinning.
1. Introduction
There is giant flux creep in high-Tc superconductors (HSTC)
due to high operating temperature. The critical current (Ic )
is defined as the current at which the electric field E induced
by flux creep reaches a criterion Ec . It is well known that
flux creep is governed by the effective barrier of flux diffusion
or pinning energy (U ) which is dependent on current density
(j ). Therefore, U is the key quantity governing Ic in the flux
creep state. It has been pointed out that U depends on the field
sweeping rate (dH /dt) in a dc magnetic ramping experiment
(U ∝ ln(1/(dH /dt))) [1], or time in magnetic relaxation
(U ∝ ln(1 + t/t0 )), or frequency (U ∝ ln(1 + 1/f t0 )) in ac
susceptibility (acs) [2]. Based on these ideas, earlier works
on the effect of driving frequency on critical current density
(jc ) by acs measurement have been undertaken [3–5]. The
3
Author to whom correspondence should be addressed.
0953-2048/01/060346+07$30.00
© 2001 IOP Publishing Ltd
underlying physics of the driving frequency dependent Ic is
the electric field induced by flux creep which has already met
the Ec when j is lower than jco , the critical current density
without flux creep. The electric transport measurement of Ic
with a different sweeping rate of applied current (dI /dt) was
carried out based on the same consideration as in [6]. It is
shown that Ic decreases with increasing dI /dt for Ag-sheathed
Bi2−x Pbx Sr2 Ca2 Cu3 O10 (Ag-Bi2223) tapes. The larger the
dI /dt, the smaller the Ic , suggesting that Ic depends on the
time scale of the experiment and is consistent with the acs
measurements [3–5]. For a given I in a V –I measurement,
different dI /dt also imply different times because t =
I /(dI /dt). Hence this result in fact indicates that flux creep
is also important in the electric transport measurement of
Ic . Unfortunately, the limited precision and ability of the
instruments results in the Ic measurement being conducted
on only a few samples and experimental parameters (dI /dt,
temperature T and applied field H ) as reported in [6]. To the
Printed in the UK
346
Effect of flux creep on Ic measurement
best of our knowledge, even such experiments as in [6] are
seldom found in the references. On the other hand, it has been
numerically shown that the experimental time interval affects
the V –I curve, apparently suggesting that a simulation study
on this subject is more convenient [7–9]. However, the detailed
role of giant flux creep in electric transport measurement of Ic
is still an open question. One of our goals in the present paper
is to study the effect of giant flux creep on Ic under different
experimental conditions (dI /dt, T , H , etc) for various samples
with different flux pinning etc, by numerically solving the flux
diffusion equation.
In the determination of Ic , the experimental V –I curve is
usually fitted in terms of a power law, V ∝ I n . It is known
that n is also an important parameter to be measured together
with Ic because n manifests the character of a sample itself.
The larger the value of n, the sharper the transition in the V –I
curve, and thus the better the quality of a sample. In fact, in
the international standard of measurement of Ic on classical
superconductors, both Ic and n are the main parameters that
have to be determined. However, the experimental n usually
depends on I at which n is determined, suggesting that either
it is not appropriate to choose n as the characteristic parameter
or the method of determining n is questionable. It turns out in
the present study that the usual method of determining n needs
to be improved, as shown below.
In this paper, numerical observation of the V –I curve is
reported, which is the basis for determining Ic by the electric
transport method. By numerically solving the electric dynamic
equation based on the collective creep model, we observe the
effect of dI /dt and other parameters on the V –I curve and
Ic . Based on this study, a suitable method to determine n is
suggested.
2. Equations
We consider an infinite slab in the y–z plane with thickness d
along the x-axis. When current is applied along the y-axis,
E and j have only y components: E(x, t) and j (x, t),
respectively, and the vortex density denoted by B(x, t) is along
the z-axis.
The electric field diffusion can be described by the
Maxwell equations
∂B
∂E
=−
∂t
∂x
∂B
∂E
= −µµ0 j − µµ0 εε0
.
∂x
∂t
For superconductors (µ 1) and normal conductors at low
frequency the second term (µµ0 εε0 (∂E/∂t)) on the right-hand
side is much smaller than the first, such that it can be neglected
[2]. Therefore, these equations can be reduced to:
∂E
1 ∂ 2E
=
∂t
g(E) ∂x 2
where g(E) = µ0 (∂j/∂E) is a function which can be
determined by a particular mechanism of flux creep. Here
we assume U = Uo ln(jco /j ) and U0 is a barrier scale [10].
U
According to E = Bv = Bv0 e− T , the electric field induced
by flux creep can be described by [11–16]:
n∗
j
U0
E = E0
(1)
n∗ =
jco
T
where E0 is the electric field at j = jco . Note that jco , U0 and
thus the exponent n∗ are functions of temperature T , magnetic
field H and pinning strength. If n∗ = 1, equation (1) reduces
to Ohm’s law, describing the normal state or flux flow state.
For infinitely large n∗ , equation (1) describes the Bean model:
E = E0 for j = jco , otherwise E = 0 for j = 0. When
1 < n∗ < ∞, equation (1) describes nonlinear flux creep.
Substituting equation (1) into the Maxwell equation, the
basic equation for numerical calculation is obtained as follows:
1/n∗
2
nE0
∂E
∗ ∂ E
E 1−1/n
.
=
∂t
µ0 jco
∂x 2
(2)
The boundary conditions are obtained from µ0 (∂/∂t)j =
d
∂ 2 E/∂x 2 and ∂/∂t 0 j dx = dI /dt:
µ0
d
dt
d
j dx = µ0
0
dI
∂E ∂E −
.
=
dt
∂x x=d
∂x x=0
Because the electric field is symmetrical at x = 0 and x = d,
the first boundary condition is
∂E µ0 dI
∂E =−
=−
.
(3)
∂x x=0
∂x x=d
2 dt
From ∂t
µ0
d/2
0
d
dt
j dx =
0
d/2
1
2
dI /dt, it yields
j dx =
∂E µ0 dI
∂E =
−
.
2 dt
∂x x=d/2
∂x x=0
So the second boundary condition is
∂E = 0.
∂x x=d/2
(4)
Equations (3) and (4) are the boundary conditions.
The initial condition is
E(x, t)|t=0 = 0.
(5)
By solving equations (2)–(5), E(x, t) is obtained. Then
according to equation (1), j (x, t) is also obtained.
To determine Ic , the
d V –I curve should be calculated first.
It is obvious that I = 0 j dx. It is assumed that V = EL,
where L is the length between the two voltage leads, E is
d
the average voltage: E = (1/d) 0 E dx. Hence, the V –I
curve can be calculated by integrating the E–j curve.
In the calculation, we used typical values of the
parameters: E0 = 0.1 V m−1 , d = 4 × 10−2 m. Since all
the quantities are normalized, the choice of parameters will
not affect the final results.
For convenience, I , j , V and x are normalized by Ico ,
jco , V0 and d/2, respectively. We should bear in mind that
the parameters Ico , jco and V0 are different from Ic , jc and
Vc , respectively. As stated above, jco is the critical current
density without flux creep, E0 is the electric field at j = jco ,
whereas Ic = jc d is an experimental quantity defined by Vc .
Correspondingly, Ico is the critical current without flux creep
and V0 is the voltage at I = Ico . Due to giant flux creep, Ico
may be much larger than Ic .
347
Y H Zhang et al
10
1.0
0.6
0.4
0.2
*
n = 10
2
jco = 1 kA/mm
10
-1
V/Vo
V/Vo
0.8
dI/dt (Ico/min):
6 0.2
3 0.12
1.2 0.03
0.6 0.027
0.3
0
10
-2
Vp
0.0
Ip
0.0
0.2
0.4
0.6
0.8
dI/dt (Ico/min): *
n = 10
6
2
jco = 1 kA/mm
3
1.2
0.6
Vp
0.3
0.2
0.12
0.03
(a)
1.0
10
(a)
-3
10
10
0.9
0.3
*
n = 50
2
jco = 10 kA/mm
10
10
Vp
0.0
Ip
0.0
0.2
0.4
0.6
0
-1
0.8
-2
dI/dt (Ico/min)
0.3
0.2
0.12
0.06
0.03
*
n = 50
2
jco = 10 kA/mm
Vp
(b)
1.0
I/Ico
Figure 1. Numerical V –I curves as a function of dI /dt.
(a) n∗ = 10. In practical measurement, such large dI /dt values as
6 Ico min−1 and 3 Ico min−1 are difficult to access. These data are
presented here only for the purpose of comparison. (b) n∗ = 50. In
practical measurement, such large dI /dt values as 0.3 Ico min−1 are
difficult to access. These data are presented here only for the
purpose of comparison.
3. Numerical simulation results
3.1. The V –I curve
Because an experimental Ic is defined as I at V = Vc , we first
calculated the V –I curves at different experimental parameters
such as T , B, dI /dt and n∗ etc. Furthermore, the effect of T ,
B and pinning strength on flux diffusion can be reduced to
n∗ and jco [8]; the V –I curves under different experimental
conditions and for different samples can be simply described
by the V –I curves at different n∗ and jco . For simplicity, only
the calculated V –I curves for n∗ = 10, jco = 1 kA mm−2
and n∗ = 50, jco = 10 kA mm−2 at different dI /dt are shown
here.
It can be found from figures 1(a) and (b) that all the V –I
curves consist of two parts converging at the characteristic
current point Ip or the corresponding Vp at a given (dI /dt)p .
The characteristic Ip and Vp depend on the current sweeping
rate dI /dt. The smaller the value of dI /dt, the smaller are Ip
and Vp . We will temporally call Ip , Vp and (dI /dt)p critical
points. In the regime where I < Ip every V –I curve deviates
from its own critical point and the separated curve is elevated
by increasing dI /dt, whereas in the regime where I > Ip , all
348
10
0
V/Vo
V/Vo
0.6
-1
I/Ico
I/Ico
dI/dt (Ico/min)
0.3
0.2
0.12
0.06
0.03
Ip
10
Ip (b)
-3
10
-1
I/Ico
10
0
Figure 2. Double logarithmic V –I curves corresponding to the V –I
curves depicted in figure 1. The slope of the curves is equal to n;
V < Vp , n ≈ 2 and V > Vp , n ≈ n∗ . Vp is the voltage at which the
two straight lines meet. (a) n∗ = 10 and (b) n∗ = 50.
the V –I curves merge into one single curve. As an example,
the position of a critical point (Ip , Vp ) is indicated by arrows
in figures 1(a) and (b).
To clearly see this characteristic of the V –I curve,
figures 1(a) and (b) are re-illustrated in figures 2(a) and (b)
in a double logarithmic manner. One can find that all the
V –I curves in either figure 2(a) or (b) consist of two straight
lines converging at Ip , Vp and (dI /dt)p . In the regime where
I < Ip , the V –I curves deviate from the single curve, and the
voltage V at a given I increases with increasing dI /dt. In the
regime where I > Ip , however, the curves merge into a single
curve. If the numerical V –I curves are fitted by the following
power law relationship
n
I
(6)
V = V0
Ico
n evidently changes with current from I < Ip to I > Ip .
According to our simulation shown in figures 1 and 2, in the
region where I < Ip all the V –I curves are parallel to each
other and n ≈ 2, i.e. the power law of equation (6) is in fact
parabolic:
2
I
dI
V < Vp .
(7)
V ∝
dt p Ico
Effect of flux creep on Ic measurement
0.10
n = 10
ii
jco =1 kA/mm
0.06
2
Ic(Ico)
0.08
Vp(Vo)
i
0.8
*
0.04
0.02
0.6
0.4
Vc(V0):
0.001
0.007
0.01
0.03
0.06
0.1
0.2
0.00
0.0
0.5
1.0
1.5
0.0
2.0
(dI/dt)p(Ico/min)
Note that this n (≈2) is independent of dI /dt and the material
parameter n∗ . Thus in the regime where I < Ip , any
experimental V –I curve cannot be used to determine the
∗
material equation E = E0 (j/jc0 )n . On the other hand, in
the region where I > Ip , the V –I curves merge into a single
curve with slope n nearly equal to n∗ , i.e.
V ∝
I
Ico
n∗
V > Vp
jco = 1 kA/mm2
10-2
10-1
100
101
dI/dt(Ico/min)
Figure 3. The dependence of Vp on (dI /dt)p at n∗ = 10. Vp is the
voltage at which the current penetrates throughout the sample.
(dI /dt)p is the dI /dt value at which the current has enough time to
penetrate (diffuse) throughout the sample at the corresponding I .
The data are obtained from figure 2(a).
10-3
n* = 10
(8)
showing the characteristic depicted by equation (1),
irrespective of the values of n∗ and jco . Thus it can be
concluded that in this region (I > Ip ), the V –I curve can
be used to determine the material equation (1).
As pointed out above, the critical parameters Ip , Vp
and (dI /dt)p depend on each other. All the simulated data
including those shown in figures 2(a) and (b) are well fitted by
dI
Vp ∝
I2
(9)
dt p p
for all n∗ and jco . The relation shown in equation (9) is
displayed in figure 3.
3.2. Ic measurement
Next we consider the influence of such a V –I characteristic
on the Ic determination. According to the definition of Ic ,
the intersects of the V –I curves with a horizontal line with
V = Vc will determine Ic as a function of dI /dt. It is easy
to see, according to the result depicted above, that Ic will be
dependent on dI /dt if Vc < Vp where the V –I curves are
dependent on dI /dt. If however Vc > Vp , where the V –I
curves are independent of dI /dt, Ic will be independent of
dI /dt.
Illustrated in figure 4 is the dependence of Ic on dI /dt at
different Vc . All the data are obtained based on figure 1(a).
In figure 4 all the Ic versus dI /dt curves are divided into
two regimes, namely (i) dI /dt < (dI /dt)p and (ii) dI /dt >
(dI /dt)p . The value of dI /dt at the boundary of regimes (i)
Figure 4. The dependence of Ic on dI /dt at different criteria (Vc )
for n∗ = 10. The data are obtained from figure 1(a). The dashed
line is the boundary of regimes (i) and (ii), where dI /dt is just
(dI /dt)p while Vc = Vp .
and (ii) is just (dI /dt)p . In regime (i), Ic is determined by the
V –I curve with Vc > Vp and independent of dI /dt. That is
∗
to say, Ic is determined by the segment V = V0 (I /Ico )n of
the V –I curves. In regime (ii) where Vc < Vp , Ic increases
with decreasing dI /dt and finally reaches the dashed line in
figure 4, as reported in [6]. That is to say, Ic is determined
by the segment V = V0 (I /Ico )2 of the V –I curves. This
is an indication that Ic measured by the pulsed method with
large dI /dt may be much smaller than that obtained by the dc
four-terminal technique in which a small dI /dt is used. On
the other hand, it also indicates that if Ic depends on dI /dt
in a practical measurement, the value of dI /dt is too large
and not suitable. In fact, the dashed line in figure 4 is also
a description of the relation between Ip , Vp and (dI /dt)p as
depicted by equation (9).
3.3. Relation of the V –I curve and current profile
To fully understand the dependence of the V –I curves
on dI /dt, the corresponding current profiles at different I
(different times) are calculated. Shown in figure 5 are the
numerical dependences of the current profile on I when the
current is applied at a fixed dI /dt for n∗ = 10. It can be
seen that as for small I (I < Ip ), the current flows near the
surface and does not penetrate the sample. When I increases,
the current diffuses toward the centre. When the current is
large enough such that I = Ip , Ip is 0.68 Ico here (n∗ = 10),
it penetrates throughout the sample and j is almost the same
from point to point. From this simulation we learn that Ip is
nothing but the penetrating current at the corresponding dI /dt.
Since the velocity of flux diffusion is limited, the
penetrated state could be reached by smaller dI /dt as well. The
j profile at fixed I and different dI /dt is calculated at n∗ = 10
as shown in figure 6, indicating how dI /dt affects the current
penetration. It can be seen that the slower the rate of the applied
current, the more deeply the current penetrates. When dI /dt is
small enough such that dI /dt < (dI /dt)p , (dI /dt)p is about
0.2 Ico min−1 here (n∗ = 10), the current has completely
penetrated the sample and j is nearly constant. This simulation
confirms that (dI /dt)p is nothing but the sweeping rate of
349
Y H Zhang et al
time increases 0.91 I
co
j/jco
91 s
68 s
0.04
V/V0
60 s
0.6 Ico
30 s
*
n = 10
jco =1 kA/mm
2
b
0.3 Ico
2
0.2
ta c d
1
dI/dt = 0.6 Ico/min
0.6
0.8
e
2
0.00
0.4
dI/dt (Ico/min)
1. 1.2
2. 0.03
0.02
jco = 1 kA/mm
0.0
a
*
n = 10
0.68 Ico
(a)
10
1.0
100
Figure 5. Numerical space and time evolution of current density
during current sweeping. x/(d/2) = 1, 0 denotes the centre and
surface of the slab, respectively.
a
b
c
d
60.5 s
j/jco
time increase
39.5 s
j/jco
*
dI/dt (Ico/min): t (s):
6
6
3
12
1.2
30
0.6
60
0.3
120
0.2
180
0.12 300
0.0
0.2
2
n = 10 jco = 1 kA/mm
30.5 s
dI/dt = 1.2 Ico/min
I = 0.6 Ico
time increases
*
n = 10
2
jco = 1 kA/mm
0.0
I = 0.6 Ico
0.4
0.2
0.4
0.6
0.8
33.5 s
(b)
1.0
x/(d/2)
0.6
0.8
1.0
x/(d/2)
Figure 6. The current density profile at fixed current I , which is
applied with different dI /dt. t indicates the time taken for the
current to diffuse.
the applied current at which the current has enough time to
penetrate (diffuse) throughout the sample at the corresponding
I.
Such a penetrated state can also be reached by voltage
relaxation as reported in [7, 9, 17, 18]. When I remains
constant, the current diffuses with increasing time, i.e. current
(voltage) relaxation. In short, at a given I , a penetrated
state can be obtained in at least two ways. One way is to
apply a current with large dI /dt to a given I and wait for a
time of relaxation. The other method is to apply a current
with a small enough dI /dt. However, are these two ways
equivalent to each other? For comparison, these two processes
are demonstrated in figure 7(a) simultaneously. It is noted that
the V –t curve shown in figure 7(a) is in fact a V –I curve
because t = I /(dI /dt). It is observed that the two curves
meet at point e, where the current and voltage in curve 2 are
about equal to those in curve 1, indicating that the same current
profile can be obtained by either of the processes at a given
current. That is to say, these two processes are equivalent to
each other. The corresponding time and space evolutions of
the current density at points a, b, c and d in curve 1 during
voltage relaxation are shown in figure 7(b).
350
1000
t(s)
x/(d/2)
Figure 7. The two ways by which the penetrating current state can
be reached. One way is to apply a current with large dI /dt and then
wait for a period of relaxation. The other way is to apply a current
with a small enough dI /dt. (a) Comparison of the two processes.
Curve 1 shows the sweeping relaxation process. When t < ta , the
current is swept; when t > ta , voltage relaxation takes place.
Curve 2 shows the process when the current is swept with small
enough dI /dt. It is seen that the two curves meet at point e where
the current and voltage at curve 2 are equal to those at curve 1,
indicating that these two processes are equivalent to each other.
(b) The corresponding time and space evolution of the current
density at a, b, c and d in curve 1 during voltage relaxation at fixed
current I .
4. Estimation of the two V –I relations
Why do the V –I curves always consist of two sections with
different power exponents? As seen above, the current density
is approximately constant despite the fact that j depends on
dI /dt within the penetrating depth δ. This suggests that
the so-called ‘sub-critical state model’ (SCSM) is a good
approximation provided spatially constant but time-dependent
j (t) is employed instead of jco [2]. Hence, SCSM is adopted in
the following estimation to demonstrate the physics underlying
the above result.
In the fully penetrated state, j d = I and I > Ip according
to SCSM. Combining the material equation (1), one directly
obtains
n∗ n∗
I
j
∝
I > Ip
(10)
V = EL ∝
jco
Ico
which of course is equation (8) obtained by the simulation.
Effect of flux creep on Ic measurement
In the partially penetrated state, the vortex density at x for a
given I is B(x) = (µ0 I /2)−µ0 j x where µ0 I /2 = B (x = 0)
and I < Ip . The magnitude of the electric field in the
y-direction can be calculated from the Faraday law:
δ
µ0 I˙
(δ − x)
Ḃ dx =
E(x) =
2
x
where Ḃ = dB/dt and I˙ = dI /dt. The voltage V = LE is
2L d/2
2L δ µ0 I˙
V =
E(x) dx =
(δ − x) dx
d 0
d 0 2
δ
µ0 I˙L
1
µ0 I˙L 2
=
δx − x 2 =
δ .
d
2
2d
0
In the SCSM, δ = B(0)/µ0 j = I /2j , which leads to
µ0 I˙L
V =
2d
I
2j
2
.
(11)
This is the parabolic relation obtained in equation (7) by our
numerical solution in the partial penetrated case.
It has been pointed out that the Bean model can be
considered as a special case for n∗ → ∞ (T = 0), which
is equivalent to the absence of flux creep. In the Bean model,
j = jco otherwise j = 0. Hence, equation (11) is also effective
for the Bean model if only j (t) is replaced by jco and thus
δ = B(0)/µ0 jco = I /2jco . That is to say, in the Bean model
V =
µ0 I˙L
2d
I
2jco
2
n∗ → ∞.
(12)
Therefore, this parabolic V –I relation in the partial penetrated
case is also expected to exist in the critical state. However,
in some cases the parabolic relation may be too small to
be monitored by a conventional voltmeter used in electric
transport measurement. Such a situation can occur with
measurements at low temperatures and weak applied fields
where both n∗ and jco are large and the Bean model is a good
approximation. A large jco causes an undetected low voltage
(Vlow ), which could be estimated according to equation (12).
On the other hand, at high temperatures or in strong applied
fields where both n∗ and jco are small, the voltage (Vhigh )
should be estimated according to equation (11) where small j
may cause a detectable voltage. That is to say, Vhigh may be
observable. In fact, according to equations (11) and (12), with
the same dI /dt and I it follows that
Vlow
=
Vhigh
j
jco
2
.
(13)
As an example, we estimate the ratio of voltage for the same
Ag-Bi2223 tape at 77 and 4.2 K with the same dI /dt and I .
Assuming j = 104 A cm−2 (77 K, 0.1 T, n∗ ≈ 5) and
jco = 106 A cm−2 (4.2 K, 0.1 T, n∗ > 60), one sees
immediately that Vlow /Vhigh = (j/jco )2 = 10−4 . By this
kind of estimation, it is easy to understand why the Vhigh –I
curve might ‘climb a mountain’ in its small I stage as measured
in [6] and presented in figure 1. The Vlow –I 2 curve on the other
hand was difficult to observe using a conventional voltmeter.
This implies that for large n∗ the V –I curve measured by a
conventional voltmeter is independent of dI /dt.
Because Vlow (for large n∗ , corresponding to low
temperature, weak applied fields and strong pinning) is very
small, Vp is also small; even for large dI /dt, the situation
of Vc > Vp is encountered frequently, resulting in Ic being
independent of dI /dt as stated above. That is to say, for a
sample with large n∗ a method with very large dI /dt such
as the pulsed method can be used to determine Ic . This is
the reason why there is no report on the dependence of Ic on
dI /dt in low-Tc superconductors even now. In contrast, for
small n∗ (high temperatures or in strong applied fields), Vhigh
and thus Vp are large and the condition that Vc < Vp is usually
satisfied, leading to the dI /dt dependent Ic as reported in [6].
In this case, a method with large dI /dt, such as the pulsed
method, may cause Ic to be underestimated. Therefore, at
high temperatures or strong applied fields the pulsed method
is not suitable for determining Ic .
5. Summary
In summary, numerical simulation was performed based on
the collective creep model to study the effect of flux creep
on Ic measurement. First we studied the effect of dI /dt
(sweeping rate of applied current) on the V –I curve by solving
the nonlinear vortex diffusion equation. It was found that the
V –I curve consists of two segments, which converge at the
penetrating current Ip and Vp at corresponding (dI /dt)p . In
the segment I < Ip , the V –I curve is parabolic, V ∝ I 2 , and
independent of the material equation, whereas in the segment
∗
I > Ip it is a power law, V ∝ I n , reflecting the material
equation. This result suggests that a suitable segment on the
V –I curve to determine the material parameter n∗ is where
I > Ip . Based on the V –I curve, it is concluded that if
dI /dt > (dI /dt)p , Ic decreases with increasing dI /dt. On
the other hand Ic is independent of dI /dt if dI /dt < (dI /dt)p ,
indicating that a dI /dt in this part is suitable to be used in Ic
measurement. The critical parameters (dI /dt)p , Vp and Ip
depend on each other as depicted by equation (9). In fact,
Ip is not necessarily very large. The penetrated state can be
reached either by applying a current with a small enough dI /dt
or by applying a current with a large dI /dt and then waiting
for a time relaxation. This parabolic V –I relation (V ∝ I 2 )
in the regime where I < Ip is expected to exist in both flux
creep state, in which V is easily measured by a conventional
voltmeter, and the critical state, in which V may be too small
to be detected. This also indicates that a method with large
dI /dt such as the pulsed method may underestimate Ic in the
case of high temperatures or strong applied fields, whereas it
will not affect Ic in the case of low temperatures, weak applied
fields and strong pinning.
Acknowledgments
This work was supported by the Ministry of Science and
Technology of China (NKBRSF-G1999-0646) and NNSFC
under no 19994016.
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