Supercond. Sci. Technol. 12 (1999) 571–576. Printed in the UK
PII: S0953-2048(99)02531-2
Effect of electric field relaxation on
the V –I curve
P Zhang, C Ren, S Y Ding†, Q Ding, F Y Lin, Y H Zhang, H Luo and
X X Yao
Department of Physics and National Laboratory of Solid State Microstructures, and
Center for Advanced Studies in Science and Technology of Microstructures, Nanjing
University, Nanjing 210093, People’s Republic of China
E-mail: cong ren@brown.ed and syding@nju.edu.cn
Received 9 March 1999
Abstract. The effects of sweeping rate of applied current (dI /dt) on the characteristic
(V –I ) and electric field relaxation (V –t) curves have been investigated numerically as well as
experimentally. The calculation is based on the nonlinear electric field diffusion equation,
while the experiment is conducted by electric transport measurements of V –I and V –t curves
on three Ag–Bi-2223 tape samples. It is found that the V –I curves shift toward smaller
current with increasing dI /dt and V decays with time apparently. We show that a certain
dI /dt causes a highly spatially varied electric field which is the natural result of the fact that
flux lines can only diffuse over barriers with certain velocity. We also show that the electric
field diffusion causes the two phenomena in the bulk sample: (1) the V –I curve is affected by
the sweeping rate of the applied current; (2) the resistance relaxes with time.
1. Introduction
The discovery of the high-temperature superconductors has
revived interest in theory and experimental methods which
study the electromagnetic response of these extreme type-II
superconductors. It has been shown that the mixed state
physics of high-Tc superconductors is so rich that a new
branch called ‘vortex matter physics’ has been established
[1]. The electromagnetic property of the vortex matter
is sensitive to the highly nonlinear part of the V –I curve
(see [2, 3] and references therein).
Thus V –I curve
measurements are among the most useful tools in the study of
electromagnetic property of high-Tc superconductors. The
electric field E is induced by thermally activated drift of
vortices driven by the Lorentz force proportional to j × B
where j and B are current density and induction, respectively.
E is proportional to the mean drift velocity v of vortices and
thus the measurements of E in transport experiments such
as R–T or V –I ones are equivalent to the measurements of
the drift v of vortices, where T and R are temperature and
resistance, respectively. Based on this concept, a step on an
R–T curve is ambiguously considered to be a sudden change
of v and thus a signal of vortex matter freezing or melting.
However, the vortices have to be in steady drift and thus are of
about the same velocity v. This condition is to be met in the
R–T measurement where the applied current and magnetic
field are constants, respectively. We will show below by
numerical calculation that j is nearly constant throughout
a sample in steady electrical transport measurement as well
as in magnetization relaxation experiments except for the
† Corresponding author.
0953-2048/99/080571+06$30.00
© 1999 IOP Publishing Ltd
initial stage. In contrast to the R–T (or magnetic relaxation)
experiments, the current I (or external field H ) is applied with
a certain rate dI /dt (or dH /dt), while the current (or electric
field) diffuses from sample surfaces toward the centre in
V –I transport (or in hysteresis loop) experiment. In this
kind of transport measurement, it is reasonable to suppose
that current density varies spatially and the vortices are in
unsteady motion. Accordingly, the vortex diffusion energy
barrier U , which depends nonlinearly on j , varies from place
to place. It is expected that the induced electric field E which
depends on current density j can be written in the form
E(j, B) = Ec exp[−U (j, B)/kT ]
(1)
and varies also from place to place in an unsteady case
and is very different from that of a steady case. In
fact Gurevich and Kupfer [4] and Brandt [5] have shown
that in magnetic relaxation magnetization decays in the
steady stage are very different from that in the unsteady
stage. In electric transport experiments the V –I curve
measurement is typically an unsteady case. As a matter
of fact, we have found experimentally that the V –I curves
for Ag–Bi2 Sr2 Ca2 Cu3 O10 (Ag–Bi-2223) tapes are affected
apparently by the sweeping rate dI /dt although the exact
physical explanation has not been given.
Another experimental finding is the voltage or resistance
relaxation in the GdBa2 Cu3 O7 thin film. In this kind of
transport measurement, the voltage or resistance decays with
time t (R–t curve) under a fixed applied current and induction.
This resistance relaxation in the film is explained by current
diffusion starting from the critical state [6, 7]. It is not clear
whether the voltage relaxation takes place in bulk samples
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P Zhang et al
such as the Ag–Bi-2223 tapes, and whether the relaxation
has the same physical origin.
In this paper, we report a numerical study on the effect of
applied current sweeping rate on electric field diffusion based
on the electric field diffusion equations. Electric transport
measurements of V –I and V –t curves on Ag–Bi-2223 tapes
were performed to verify the calculation results.
2. Nonlinear electric field diffusion in
superconductors
Flux creep in superconductors can be formulated in terms of
a nonlinear electric field diffusion in a sample. To simplify
the analysis of flux diffusion, various assumptions are often
made. What we consider in this paper is a slab of infinite
length along y and z axes with width w along the x axis.
When a current is applied along the y axis the electric field
E (r ) and current density j (r ) have only y components
E(x, t) and j (x, t), respectively. Thus the induced magnetic
field H is only along the z axis. In this parallel field, the
electric field diffusion can be described by the Maxwell
equations
∂B
∂E
=−
(2)
∂t
∂x
∂H
= −j.
(3)
∂x
For high-κ superconductors and B > µ0 Hc1 , where Hc1
is the lower critical field, we can put µ0 H = B. These
diffusion equations can be written in terms of different
variables H (r , t), j (r , t) or E (r , t) [8]. Here we use the
latter representation since it turns out that the time evolution
of E (r , t) is universal for different models of thermally
activated flux creep [4]. Thus equations (2) and (3) reduce to
(4)
∂j
(5)
∂E
where the function g(E) is determined by the particular
mechanism of flux creep, namely, the U (j ) relationship:
g(E) = µ0
µ0 kT
g(E) = −
E
∂U
∂j
−1
.
(6)
Nonlinear flux dynamics described by equation (4) or a
similar equation for H (x, t) have widely been studied for
different flux creep models [9–11]. Here we pay attention
to the logarithmic potential barrier U (j ) = U0 ln(jc /j )
where U0 is a characteristic barrier scale [8, 12]. This U (j )
relationship leads to the well known power dependence of
the E–j curve according to equation (1)
n
j
E = Ec
n = U0 /kT .
(7)
jc
It is noted that equation (7) can describe ohmic behaviour,
for which n = 1, nonlinear flux creep, which corresponds to
n > 1, and the Bean critical state, n → ∞. Combined with
572
1/n
∂E
nEc
∂ 2E
=
E 1−1/n 2 .
∂t
µ0 jc
∂x
(8)
This is the basic equation describing the diffusion of the
electrical field.
2.2. The boundary and initial conditions
2.1. Basic equations
∂ 2E
∂E
= g(E)−1 2
∂t
∂ x
equations (5) and (7), the electric field diffusion equation (4)
can be rewritten as
In V –I curve measurements, current is always applied with
a sweeping rate. That is one measures the voltage V while
the current is increased with a rate dI /dt. So we obtain
Z w
dI
.
j dx =
∂t
dt
0
For symmetry,
∂E ∂E =
−
∂x x=0
∂x x=w
(9)
according to the Maxwell equations (2) and (3),
µ0 ∂t j =
we obtain
Z
w
µ 0 ∂t
0
so
∂ 2E
∂x 2
(10)
∂E(x, t) ∂E(x, t) j dx =
−
.
∂x x=w
∂x x=0
µ0 dI
∂E(x, t) .
=−
∂x x=0
2 dt
(11)
Furthermore, in the centre of the bulk sample, B|x=w/2 ≡ 0,
then
∂B ∂E =
−
= 0.
(12)
∂x x=w/2
∂t x=w/2
We would like to emphasize that this boundary condition here
is different from that of the critical-state model in which the
magnitude of j equals the critical current density jc [7, 13].
As for the initial condition, there is no electric field and
current at t = 0 in the sample:
E(x, t)|t=0 = 0.
(13)
In our calculation, we assume the parameter n = U0 /kT
is independent of dI /dt, whereas it is dependent on flux
pinning strength, magnetic field and temperature. jc is
the critical current density defined at a certain electric field
criterion Ec = Vc /d with Vc a voltage criterion and d the
length between measuring points of the voltage. In this
work, we employ the typical magnitudes jc = 1010 A m−2 ,
Ec = 1 µV cm−1 and w = 10−3 m.
3. Numerical results and discussions
3.1. The effect of current sweeping rate dI /dt
In figures 1(a) and 1(b) we plot the typical numerical results
of the effects of the sweeping rate of applied current on
the distributions of the electric field and current density at
Effect of electric field relaxation on the V –I curve
t = 10 s with n = 8, respectively. It is apparent that both
E and j vary from place to place. At a given position the
higher the sweeping rate of the applied current, the higher
the electric field and current density. For a given sweeping
rate, the electric field E is lower at the position away from
the surface of the sample than that at the surface. It is
worth mentioning that for the unsteady case, this kind of
spatial distribution is typical for either E or j and completely
different from that of the critical state model or steady cases.
For example, in the magnetic relaxation (steady case), the
constant current density j (the Bean critical state model)
is a good approximation [14]. Shown in figures 2(a) and
2(b) are the distributions of E and j at different times
(i.e. different I , I = (dI /dt)t) under the sweeping rate
dI /dt = 100 A min−1 , respectively. At t = 0 there are
no electric field and current in the sample. As time increases
both E and j diffuse gradually into the sample. At t = 100 s
(equivalent to t → ∞) the electric field has penetrated fully
into the sample, and the current density j is almost the same
value through the sample.
In order to make the above current and field measurable,
we now calculate V –I curve using the above E(x, t) and
j (x, t). It should be borne in mind that one can only measure
a kind of average electric field between the two electric
contacts in a V –I experiment. Here we assume the measured
voltage V is a simple average of the position dependent
electric field E(x, t):
V (t) =
2d
w
Z
(a)
w/2
E(x, t) dx.
(14)
0
Combining the data shown in figures 1 and 2 and equation
(14), we obtain V –I curves at different dI /dt. Displayed in
figure 3 are the numerical results. It is clear that the voltage
depends on dI /dt and is higher under higher sweeping rate
than lower sweeping rate for a given current. Accordingly,
the V –I curves move toward smaller current with increasing
dI /dt. This is the main result of our numerical calculation.
The fact that the magnitude of V decreases with decreasing
sweeping rate for a fixed I implies that there exists electric
field relaxation, which will be discussed in detail below. Here
it is pointed out that any measured V –I characteristic curve
for superconductors depends on not only its intrinsic and
geometric properties but also dI /dt. Hence, whenever one
compares their transport characters in terms of V –I curves
it should be borne in mind that only those V –I curves with
the same dI /dt can be used, otherwise, the comparison is
meaningless. Unfortunately, experimental V –I curves have
been reported in various references without indicating dI /dt.
The numerical calculation of the influence of the
parameter n has also been made. Illustrated in figure 3 is one
of the typical results. It is found that the larger n, the sharper
the transition of the V –I curve is. This is in accordance
with a large number of experimental facts. These results are
easy to understand because n is proportional to U0 /T , and
U0 is actually a function of magnetic field, temperature and
flux pinning. Thus n is a parameter reflecting the role of
temperature, magnetic field and pinning. Different n implies
different effective pinning.
(b)
Figure 1. The distribution of the electrical field E(x) and the
current density j (x) under various sweeping rates of applied
current dI /dt with t = 10 s and n = 8. (a) E(x); (b) j (x).
3.2. Relaxation of the electric field
Now we explore the relationship between the dI /dt
dependent V –I curve and the voltage (electric field)
relaxation V –t curve for the bulk sample. We have called
the former an unsteady case and the latter a steady one. In
the electric field relaxation, current I (t 0 < 0) is applied first
with a constant dI /dt until I (t 0 = 0) is reached, then the
I (0) is fixed while the field E(x, t 0 ) decays. This kind
of electric field relaxation has been studied theoretically
and experimentally for film samples [4–7, 15]. Clearly, the
relaxation is still described by equation (8). The initial
conditions of the relaxation j (x, t 0 = 0), E(x, t 0 = 0) can be
obtained by taking j [x, t = I /(dI /dt)], E[x, t = I /dI /dt)]
calculated in the last section as shown in figure 2. From
now on, the symbol 0 will be omitted and we let t = t 0
573
P Zhang et al
(a)
(a)
(b)
(b)
Figure 2. The distributions of electrical field and current density
at different applied currents, i.e., time, respectively, with
dI /dt = 60 A min−1 . (a) E against x; (b) j against x.
for simplicity. It is noted that the difference between our
calculation and that reported in the references is only the
initial conditions. In the references, either E = 0 or a critical
state distribution is used. Shown in figure 4 are the local
field evolution curves which are very similar to the reported
experimental and theoretical relaxation curves R(t) at t t0
described by Zeng et al and Brandt and Gurevich [5, 7, 8] (t0
is the relaxation time scale which depends on the sweeping
rate of magnetic field dH /dt).
Shown in figure 5 are the calculated electric field
relaxation curves, which can be detected experimentally.
These numerical curves for the bulk sample are very similar
to those measured by Ma et al [6] for a film and calculated
by Brandt [5] and Zeng et al [7]. Figure 5 shows also that
the power n has an important influence on the relaxation
process. The increase of n decreases the relaxation rate.
574
Figure 3. The calculated characteristic V –I curves under
different sweeping rates of current. (a) n = 8; (b) n = 20.
As pointed out above, the reason is that n reflects the
effective pinning strength governed by temperature, magnetic
field and quenched disorder. The electric field diffusion
process is retarded at high energy barrier or low temperature.
Thus our calculation predicts that there exists resistance
relaxation in the bulk sample as well. Furthermore, the two
physical phenomena, i.e. the V –I curve depends on dI /dt
and the resistance decays with time, are calculated by the
same equation (i.e. equation (8)), and thus are of the same
mechanism: highly nonlinear electric field diffusion.
4. Experimental evidence
To verify the above numerical prediction that any bulk
samples with a dI /dt dependent V –I curve will exhibit
resistance relaxation, we have carried out electric transport
measurements on both the V –I –dI /dt and V –t curves for
Effect of electric field relaxation on the V –I curve
Figure 4. The time and space evolution of the electric field,
resulting in the resistance relaxation.
(a)
Figure 5. The calculated electric field relaxation curves for
different parameters n, which reflect the influence of temperature
T , magnetic field H and flux pinning strength.
three types of Ag–Bi-2223 tape (BPK, BRK, HAG). These
samples have approximately the same size including their
outer silver sheath and the superconductor, i.e. 30 × 5 ×
0.25 mm3 . The transport measurements by the usual fourprobe method are carried out with the sample immersed
in liquid nitrogen. Contacts to the sample were made by
soldering leads to the Ag sheath. The distance between the
voltage contacts is 10 mm. The details of the samples used
in this experiment have been reported in a recently published
paper [16].
Figure 6 shows the typical experimental V –I curves as
a function of sweeping rate of applied current for two of the
tapes (BPK and BRF). These V –I curves are very similar
to that reported in [16] and the calculated curves shown in
figure 3. Because the length between the two voltage contacts
is 10 mm, the voltage V is in fact the electric field E, i.e. V =
E (µV cm−1 ). It is apparent that the V –I curves are shifted
toward smaller currents with increasing dI /dt. This kind of
(b)
Figure 6. The experimental V –I curves at different applied
current sweeping rates for Ag–Bi-2223 tapes. (a) BPK; (b) BRF.
V –I curve is also obtained for tape HAG, but not shown here
for simplicity. It is interesting that these V –I curves have
implied the resistance relaxation. For example, in figure 8(a),
I = 24 A, V = 14 µV for dI /dt = 100 A min−1 ,
V = µV for dI /dt = 60 A min−1 , V = 3.4 µV for
dI /dt = 20 A min−1 , V = 3 µV for dI /dt = 100 A min−1
and V = 2.2 µV for dI /dt = 4 A min−1 . That is to say,
t (V = 14 µV) = 14.4 s, t = (V = 8 µV) = 24 s,
t (V = 3.4 µV) = 72 s, t (V = 3 µV) = 144 s and
t (V = 2.2 µV) = 360 s. These data indicate the relaxation
of the voltage or the electric resistance.
Typical measured relaxation curves of the electric field
575
P Zhang et al
at a given current V decreasing with decreasing dI /dt. When
dI /dt = 0 (keeping current constant) the electrical field still
diffuses but is more and more steady, resulting in the electric
field decay with time. Electric transport measurements of
V –I and V –t curves on three Ag–Bi-2223 tapes are in good
agreement with the calculated ones. Thus it is concluded that
the current sweeping rate dependence of the V –I curve and
the electric field relaxation have the same physical origin:
highly nonlinear electric field diffusion.
Acknowledgments
One of the authors (C Ren) gives thanks to Motorona
Co. Ltd of Beijing for the Motorona Fellowship. This
work is supported by the National Centre for R & D
on Superconductivity under contract No J-A-4102 and
the Chinese Natural Science Foundation under contract
No 19574025.
Figure 7. The relaxation of the electrical field measured in
Ag–Bi-2223 tape (BPK) in three applied currents.
are shown in figure 7 where we plot V versus time t under
three different fixed currents for the same sample (BPK). As
we can see, the voltage does relax when the current retains the
constant value. The higher the current, the faster the voltage
relaxes. These experimental results (figure 7) are similar to
the calculated ones shown in figure 5 because the electric field
relaxation is equivalent to the resistance relaxation discussed
in [7]. The experimental results shown in figures 6 and 7
not only confirm the numerical results but also demonstrate
that the two phenomena of the resistance relaxation and the
dependency of V –I curve on dI /dt result from the same
mechanism.
5. Conclusions
The electric field diffusion in unsteady and steady cases
is investigated numerically and experimentally for a
superconducting slab. In the unsteady case, i.e. in the case
of the V –I curve measurement with given sweeping rate
of applied current dI /dt, the highly nonlinear electric field
diffusion causes the dependence of electric field on dI /dt
spatially and temporally, resulting in the V –I curves shifting
toward smaller current with increasing dI /dt at given V or
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