Supercond. Sci. Technol. 12 (1999) 219–225. Printed in the UK
PII: S0953-2048(99)00023-8
Numerical simulation of flux creep in
high-Tc superconductors
A N Lykov
P N Lebedev Physical Institute of RAS, Leninsky Pr. 53, 117924 Moscow, Russia
Received 9 December 1998
Abstract. Anderson theory is modified to explain features of the flux creep in high-Tc
superconductors. The approach is based on a consideration of single-vortex pinning and
creep in a washboard potential with the presence of a normal pinning strength distribution.
It explains the scaling behaviour of the E–J curves, the logarithmic shape of the potential
barrier as a function of the transport current, the decrease of the barrier as the temperature
goes to zero and other results of the studies of flux creep in high-Tc superconductors.
Moreover, the method provides the possibility of estimating the main pinning parameters of
superconductors by fitting the calculated and experimental data.
1. Introduction
Considerable effort has gone into the study of flux creep
in high-temperature superconductors, which show many
unexpected features. The most interesting experimental
results in this field are the following.
The first
is the logarithmic divergence of the current-dependent
potential barrier U (J ) ∼ log(J0 /J ) derived from resistive
characteristics of YBa2 Cu3 O7−x films [1], where J0 is
constant, and confirmed by magnetic relaxation experiments
[2]. The second is the scaling behaviour of the electric field
as a function of current density (the E–J curve), which
is accompanied by the collapse of the log E–log J curves
into two curves with different signs of curvature [3, 4].
The third is the decrease of the apparent pinning barrier
as the temperature goes to zero [5, 6]. This cannot be
explained in the framework of the conventional Anderson
theory for flux creep [7], which assumes a thermal activation
mechanism and uncorrelated motion of vortex bundles. In
this theory, the curves always have positive curvature, and
the potential barrier depends linearly on the applied current.
The exact nature of these features is still under discussion.
All models can partially explain some experimental data,
but all have some problems. For example, a model for the
phase transition between vortex liquid and vortex glass states
was proposed [3, 4] to explain the behaviour of the E–J
curves. This approach complicates the study of pinning in
superconductors, and cannot explain the logarithmic shape of
the current-dependent potential barrier. Both this model and
the theory of collective flux creep [8] predict a power-law
dependence for the activation barrier U (J ) ∼ J α , where
α < 0. On the other hand, in spite of the large effort
put into attempting to modify the Anderson model [9–13],
this approach cannot also explain these effects. The change
of the sign of the curvature of the log E–log J curves was
explained in different models, but the values of the calculated
critical exponents significantly differ from the experimental
0953-2048/99/040219+07$19.50
© 1999 IOP Publishing Ltd
ones. Moreover, these models cannot explain the logarithmic
shape of the potential barrier and other results of magnetic
studies of flux creep in high-Tc superconductors. This is
the main reason for the application of collective pinning
theories to explain these phenomena. In this paper, we have
modified Anderson theory in order to explain these features
of flux creep in high-Tc superconductors using well known
flux-dynamics equations and traditional ideas of mixed-state
theory without introducing new concepts.
2. Modelling
Following the classical ideas of Anderson theory, our
approach is based on a consideration of the uncorrelated
interaction of isolated vortices with a modified washboard
potential. This approach is possible in the case of a
strong intrinsic pinning potential and a sufficiently weak
magnetic field. We restrict our consideration to thin films
in a transverse magnetic field. It suggests two-dimensional
computer simulation. The transport current flows parallel
to the grooves of the harmonic pinning potential. Unlike
an ordinary harmonic one-dimensional potential, we assume
that their amplitudes (Uil ) are distributed in space (see
figure 1), because the flux pinning strength is widely
distributed in practical high-Tc superconductors. In the
model the sample is divided into equal rectangles, and the
subscripts, ‘i and j ’, define their positions along width and
length of the film, respectively. The period of every single
sinusoidal pinning potential is equal to the coherence length
(ξ(T )). This is close to the experimental situation, because
the smallest transverse length scale that can be resolved
by the vortex core is the coherence length. Moreover, the
dimension of the most effective pinning centre should also
equal ξ(T ). A normal distribution for the amplitudes of
the single sinusoidal pinning potential was assumed in every
219
A N Lykov
in the creep-free case. Hence the pinning potential can be
expressed in terms of the Jcil . In our case, we can divide the
pinning centres into two types for every transport current
density (J ). There are strong pinning centres, when Jcil
is greater than J , and weak pinning centres, when Jcil is
smaller than J . In the first case, the vortex system enters
the flux-creep regime. It is obvious that the transport current
suppresses Uil [14, 15], so that for the sinusoidal pinning
potential, its current dependence is:
Uil (j ) = Uil (0)[(1 − j 2 )0.5 − j cos−1 j ]
Figure 1. Schematic representation of a modified washboard
potential without transport current (upper curve) and with
transport current (lower curve).
channel or fragment (l) of the film along the length:
(Uil (0) − U0l )2
Nil = N0l exp −
2σl2
(1)
where U0l is the most probable potential, and σl2 is a
constant representing the degree of deviation in fragment
l. The distribution parameters are chosen to obtain the
best agreement with experiment. The σl were assumed to
be a constant part of U0l for different channels. N0l is a
constant determined by the condition of normalization so
that the total number of pinning centres in the fragment
l is equal to w/ξ(T ), where w is the width of the film.
In real superconductors, different fragments of the sample
along its length do not have identical superconducting
properties, because there are many spatial inhomogeneities
in superconducting films. As a result, first an electric field
arises in the fragment with the smallest critical current, and
the part of the superconductor in a resistive state increases
with increasing transport current. To take into account
this phenomenon, we believe that the fragments along the
transport current are distinguished by their parameter U0l .
A normal distribution for N(U0l ) was adopted for the
calculation.
Each amplitude of the potential can be associated with
its own critical current density (Jcil ), since Fpil ∼ ∇Uil ,
where Fpil is the pinning force per unit length arising from the
interaction of the vortex lattice with the washboard potential,
and Jcil is the virtual critical current density of the part of
the film with a usual single sinusoidal pinning potential (Uil )
220
(2)
where j = J /Jcil . The equation is well approximated by
U (j ) ∼ (1 − j )1.5 when j → 1. Thermal activation
results in a hopping motion of the vortices, which leads to
an electric field. The flux creep gives in the case of a uniform
washboard potential (without amplitude distribution) the
following expression for the induced electric field:
−π Uil (0)J
−Uil (j )
1
−
exp
=
Bξ(T
)
exp
Ecil
kB T
kB T
(3)
where kB is Boltzmann’s constant, B is the magnetic
induction and  is the depinning attempt frequency with
which vortices try to escape from the pinning well. At
present,  is unknown in detail, but is assumed to lie
in the range 103 –1011 Hz [16, 17]. For example,  was
estimated by Brandt as the characteristic frequency of the
thermal fluctuations of a vortex lattice [11]. Another
mechanism for vortex excitation in pins is based on the
following consideration. Vortices located at strong pinning
centres interact with neighbouring fast-moving vortices. The
flux motion in the part of the superconductor with weak
pinning potential creates an oscillating electromagnetic field
that excites the pinned vortices. In real superconductors,
every vortex interacts with many other vortices, due to their
wandering [18]. Therefore, the movement of even a few
vortices is of great importance for the excitation of the vortex
system. This mechanism offers a way of taking into account
mutual interaction of the vortices in the Anderson model.
Since E = Bv, where v is the vortex velocity, equation (3)
leads to the time spent by a vortex in one strong pinning centre
−π Uil (0)J −1
Uil (j )
1−
. (4)
τcil = −1 exp
kB T
kB T
For large current or weak pinning centres, when J > Jcil ,
the vortex system enters a flux flow regime. The equation of
motion of every vortex line in the washboard potential can be
written
(5)
ηv = FL − Fpil
where η is the viscous damping coefficient of the flux motion
and FL is the Lorenz driving force corresponding to a uniform
transport current density. As usual, we do not take into
account the inertial component in this equation, which is
significantly smaller than the viscous drag force ηv. For
a uniform washboard pinning potential, this equation is very
similar to that for the time-dependent phase in a resistively
shunted Josephson junction [19]. As a result, equation (5)
yields an oscillating electric field, and its time average is
given by
2 0.5
)
(6)
Eil = ρf (J 2 − Jcil
Flux creep in high-Tc superconductors
where ρf is the flux flow resistivity. In this case, the time for
a vortex to move over one pinning centre with dimensions
equal to ξ(T ) is given by
2 −0.5
τf il = ξ(T )η(J 2 − Jcil
) /80
(7)
where 80 is the magnetic flux quantum.
The total time τl spent by a vortex in channel l of the film
is determined by the time spent in strong pinning centres and
by the time for the viscous flux motion in the remaining part
of the sample. We can find this time taking into account the
normal distribution for the amplitudes of the single sinusoidal
pinning potential
X
X
τl =
Nil τf il +
Nil τcil
i
i
where the first sum is over all weak pinning centres and strong
pinning centres in channel l, respectively. As a result, the
electrical field can be written as:
El = Bw/τl .
(8)
Finally, we should sum the electric field of all fragments to
find the E–J curves of the sample:
X
E=
Nl El .
(9)
l
This method for calculation of the E–J curves has
difficulties taking into account pinning centres with Jcij near
J , since equation (7) yields τf il → ∞. That is not real,
since the maximum time spent by a vortex at a weak pinning
centre in the flux-flow regime is restricted by hopping motion
of the vortex and equal to −1 . To overcome this problem we
exchanged the τf ij resulting from (7) when τf ij > −1 with
−1 in our program. Thus, flux motion is transformed into
hopping motion for small electric fields. The pinning strength
distribution makes it possible to decrease the calculation error
even by using this simple technique, thinking that the time
spent by the vortices at these centres is a small fraction the
τj . In our case, the additional error in the calculation of the
E–J curves is a few per cent.
The Maxwell equation ∂ B /∂t = −rot E is used to find
the relaxation of the magnetization or current density. For a
thin superconducting hollow cylinder with the characteristic
dimension, L, under a parallel magnetic field, this equation
gives approximately
E(J ) = −∂(hBiL)/∂t = −(µ0 L2 /2)∂J /∂t
(10)
where hBi is a mean value of the magnetic flux density, and
E(J ) is defined by relation (9). Equation (10) is solved
numerically in our work with initial condition: J (t = 0)
is equal to the critical current of the sample.
3. Results of calculations and discussion
The calculated E–J curves for the sample reported in [3] and
[4] are plotted in a double logarithmic scale in figure 2(a).
We have assumed typical values of the parameters for
YBa2 Cu3 O7−x films. If, in accordance with the experiment,
we restrict the voltage range (−1 < log10 E < 2), the
qualitative agreement between the series of curves presented
in [3] and in figure 2(a) becomes evident. Best agreement
with experiment was achieved for  = 1.5 × 109 Hz. The
curvature of the E–J curves changes sign at T = 77.5 K,
which corresponds to the melting point (Tg ) in the model of
the liquid-glass vortex state transition. In our case, this is
the transition from flux flow to flux creep in the investigated
voltage region, and no changes of state occur. At low
temperatures, the E–J curves are approximated by flux-flow
relation (6), which gives negative curvature. In contrast, at
high temperatures, the curves are approximated by flux-creep
relation (3), which gives positive curvature. It will be noted
that this experimental result is not explained by the Griessen
model [9], which also incorporates a distribution of activation
energies but does not take into account the space shape of the
pinning centres. Moreover, our model gives a power-law
E–J curve for T = Tg in this voltage range.
Finally, our model can also explain the scaling of
the experimental E–J curves presented in [4], where,
in accordance with this work, the scaled resistance
(E|T − Tg |γ (1−z) /J ) is plotted against the scaled current
J /|T − Tg |−2γ , where z and γ are the dynamic and static
critical indices in the theory of the vortex glass–liquid phase
transition. The collapsed E–J curves calculated using our
model are shown in figure 2(b). Note that the model makes
it possible to change the slope of an E–J curve plotted in a
double logarithmic scale at T = 77.5 K in a wide range by
varying the parameters of the sample and spatial distribution
of the pinning potential. As a result, agreement with the
experimental slope can be achieved. This makes it possible
to achieve equality between the experimental and calculated
values of z, which is the linear function of the slope of the
log E–log J curve at Tg . In figure 2(b), z = 4.8, in agreement
with the experiment [3, 4].
A difficulty arises when we try to fit the theoretical
and experimental values of the second critical exponent γ .
Usually, the E–J curves from the model collapse into a single
curve with γ ≈ 1. Thus, it is very difficult to obtain precise
scaling of the E–J curves for γ equal to the experimental
value (1.7) for current-independent . For example, the best
collapse of the E–J curves calculated with parameters typical
for YBa2 Cu3 O7−x films occurs with γ = 1.1, as shown in
figure 2(b). Similar collapsed E–J curves were also obtained
for Brandt’s dependence of (T , B). While a γ value that
was approximately equal to 1 was found in some experiments
[20–22], the problem of the agreement of the γ with the
experiment [4] remains. The agreement can be achieved
taking into account the mutual interaction of the vortices,
which results in the excitation of the vortices located at
strong pinning centres by neighbouring fast-moving vortices.
In this case, the characteristic or maximum frequency of
the excitation, which is the depinning attempt frequency,
is proportional to the transport current in agreement with
equation (5):
(11)
 = 80 J /ηξ.
On the other hand, we believed that the vortex interaction
is too weak to have noticeable influence on the pinning
potential. The E–J curves and collapsed curves calculated
using this relation for  are plotted in figure 3(a) and
(b). In this case, we believed U0 (T ) = U0 (0)(1 − (T /Tc )2 )
221
A N Lykov
(a)
(a)
(b)
(b)
Figure 2. (a) E–J curves calculated with  = 1.5 × 109 Hz for
the sample reported in [3]. The temperature ranges from 75.5 to
79.5 K in 2 K intervals and H = 4 T. (b) The collapsed E–J
curves calculated using the scaling forms reported in the text with
γ = 1.1, z = 4.8 and Tg = 77.5 K. The temperature ranges from
74.5 to 79.5 K in 0.1 intervals.
(1 − (T /Tc )4 )0.5 , U0 /kB = 8000 K and σ = 0.5U0 . The
most probable critical current density Jc0 is assumed to have
the following temperature dependence: Jc0 (T ) = Jc0 (0)
(1 − (T /Tc )2 )1.5 , where Jc0 (0) = 5×109 A m−2 . Evidently,
the quality of our collapse is good enough. Here, in
agreement with experiment [3, 4], z = 4.8 and γ = 1.7.
Thus, the model explains not only the scaling behaviour of
the E–J curves but also the log E–log J collapse. Since we
consider the case of thin superconducting films, the model
222
Figure 3. (a) The E–J curves calculated for the same
temperatures as for the curves in figure 2(a) using the current
dependence of . (b) The collapsed E–J curves calculated using
the scaling forms with γ = 1.7, z = 4.8 and Tg = 77.5 K.
explains also the similar properties of the E–J curves in this
case. Firstly, the critical scaling behaviour of the E–J curves
in YBa2 Cu3 O7−x thin films was found by Sawa et al [23].
It cannot be explained by a model with a phase transition
between vortex liquid and vortex glass states, which is applied
only to a bulk superconductor or to a thick superconducting
films [3].
Using the value of the most probable pinning energy U0
estimated above, we can check the applicability of the single
vortex approach to the analysed case. It is well known that
Flux creep in high-Tc superconductors
Figure 5. Logarithmic current dependence of the apparent
activation energy obtained using our calculations.
Figure 4. Calculated magnetic-field dependence of Tg .
neighbouring flux lines can be bound together into bundles
by the interaction of their fields. The value of the bundles
depends on the pinning forces: the more the pinning the less
the quantity of the vortices in the bundle. In the limit of very
strong pinning, the single vortex approach can be applied.
The value of the most probable bundle can be obtained by
using collective pinning theory [24]. In this theory, the
pinning potential for the flux bundle is written as
2
/µ20 B)1/4
U0 = 0.643(g 2 /ζ 3/2 )(870 Jc0
where g2 is the number of fluxoids in the flux bundle, µ0
is magnetic permeability of free space and ζ ∼ 15. This
relation gives g 2 ≈ 2 at T = 77.5 K. The small number of
the vortices in the flux bundle means that the elastic energy
of the vortices is negligibly smaller than the pinning energy.
Thus the bundle size is small enough to use the single vortex
approach.
The model enables us to explain the decrease of Tg
with increasing B [3]. In our case, Tg is determined by the
temperature at which the curvature of the E–J curve changes
from concave to convex. The calculated dependences are
similar to the experimental ones: see figure 4. The Tg
depends on U0 . The larger U0 , the larger Tg , since the
temperature interval where the role of the hopping motion is
large in comparison with the flux flow motion increases. As a
result, the transition from flux flow to flux creep should occur
at a higher temperature in some electric field window since
the pinning energy is suppressed by an external magnetic
field.
On the other hand, if the current dependence of the
activation energy is analysed at a constant temperature, the
influence of (J ) on the E–J curves is equivalent at small
current to a logarithmic current dependence of the activation
energy in models with constant , since
(J /J0 ) exp{−Uil /kB T } = exp{−[kB T ln(J0 /J )+Uil ]/kB T }.
(12)
It gives a logarithmic current dependence of the apparent
activation energy at every temperature in accordance with
the experiment [1, 2]. This is supported by our calculations.
Figure 5 shows the current dependence of the apparent
activation energy derived from the model. This dependence
was obtained using the following method. First, E(J )
was calculated for a current J using equation (11) for the
attempt frequency. Then by calculating the E–J curves
with a current-independent attempt frequency, a new U0 was
selected to give an E(J ) equal to that first obtained. In
other words, the E–J curves, calculated using relation (11),
are analysed in the framework of the flux-creep equations
with current-independent , as carried out in the experiments
[1, 2, 16]. Figure 5 shows the logarithmic current dependence
over a wide range. Thus, this mechanism for vortex excitation
enables us to obtain the origin of the logarithmic current
dependence of the apparent activation energy, which makes
it possible to explain, for example, the quasi-exponential
behaviour of the measured Jc (T ) dependence [16, 25]. The
role of this mechanism for the excitation of pinned vortices
reduces with decreasing transport current, since the vortices
spend more time at the pinning centre and do not move. This
increases the role of vortex–lattice vibration. At some J , the
first mechanism is replaced by the second, and U0 (J ) will
approach a constant at small transport currents. A similar
current dependence for the activation energy was found in
YBa2 Cu3 O7−x films [26], giving additional support for this
model.
Moreover, the analyses of the calculated J (t)
dependences enable us to explain the decrease of
the apparent activation energy U0∗ with lowering the
temperature.
The analyses is based on the linear
approximation J (t) = Jc (1 − (kB T /U0 )) ln(t/t0 ), where t0
is a characteristic time, of the real dependences, which deviate
from the logarithmic type. In this case, the J (t) dependences
are found by calculating equation (10). In agreement with
experiment [5, 6], the mean creep rate h∂j/∂ ln ti in a certain
time ‘window’ gives kB T /U0∗ , and the U0∗ decreases as the
223
A N Lykov
Figure 6. Real (U0 ) and apparent (U0∗ ) pinning potential depth
against temperature.
temperature goes to zero, as shown in figure 6. At the same
time, the pinning potential U0 (T ) increases with decreasing
T . There are two origins of this anomalous dependency
in our model. The first is the nonlinear dependency (with
positive curvature) of the pinning energy as a function of
current (2). The value of the J , around which the analysis is
centred in the time ‘window’, decreases as the temperature
increases. Therefore, it results in decreasing Uapp obtained
by extrapolating the tangent to the U (j ) curve at a given value
of J to J = 0 with decreasing T . A similar explanation of this
effect was found earlier by Welch [27], and by Matsushita and
Otabe [15]. The second is the influence of the (J ), which
results in the logarithmic term to the activation energy in the
usual models with constant . Evidently, the term, which
has the additional coefficient kB T decreases with decreasing
T.
Using this approach, we can find also the temperature
dependency of the irreversibility field Birr (T ). This field is
defined in the magnetic field–temperature plane as a plane
boundary between zero and nonzero critical current density.
In our calculations, the Birr is defined by the magnetic field
at which the Jc is reduced to 1 × 105 A m−2 . Figure 7 shows
the Birr (T ) curve derived from the model. It is empirically
known [27] that the Birr varies with temperature in Y-based
superconductors as
T 3/2
Birr (T ) ≈ 1 −
.
Tc
(13)
This dependency is shown in figure 7 by a line. The curve
obtained using our model is close to the curve calculated using
relation (13). Here, we do not analyse the Birr (T ) curves near
the critical temperature, where the approach does not work.
The origin of the irreversibility line in our case is depinning
of fluxoids. As a result the Birr increases with increasing U0 .
224
Figure 7. Irreversibility lines calculated using our model (points)
and empirical equation (13) (line).
4. Conclusions
In this work, we found that both the transport current
and magnetic field features of high-Tc superconductors can
be explained by thermally activated flux creep of isolated
vortices for a modified washboard pinning potential. Our
model incorporates in the Anderson theory flux flow, a
distribution of activation energies, the mutual interaction
of the vortices and shape of the pinning centres. Thus,
the model is only an attempt to approach the real situation.
Quantitative agreement between the model and experimental
data for various high-Tc superconductors can be obtained.
The method makes it possible to estimate the main pinning
parameters of superconductors by fitting the calculated and
experimental data. The irreversibility field was found to
depend strongly on the flux pinning strength so it can be
increased by improvement of the vortex pinning.
Acknowledgments
This work was supported by the Russian Scientific Council
NSTP on Condensed Matter, ‘Priority Areas in Condensed
Matter Physics’, (grant No 96041), and the Russian
Foundation for Basic Research (grant No 97-02-17545).
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