Supercond. Sci. Technol. 13 (2000) 356–361. Printed in the UK
PII: S0953-2048(00)09495-1
Current distribution and ac
susceptibility response of a thin
superconducting disc in an axial field:
a theoretical approach
S A Aruna, P Zhang, F Y Lin, S Y Ding and X X Yao
Department of Physics and National Laboratory of Solid State Microstructures,
Centre for Advanced Studies in Science and Technology of Microstructures,
Nanjing University, Nanjing 210093, People’s Republic of China
Received 12 November 1999
Abstract. Within the framework of the thermally activated process of the flux line or flux
line bundles, and by time integration of the 1D equation of motion of the circulating current
density J(ρ, t), which is suitable for thin superconducting films (R d, λ), we present
numerical calculations of the current profiles, magnetization hysteresis loops and ac
susceptibility χn = χn + iχn for n = 1, 3 and 5 of a thin disc immersed in an axial
time-dependent external magnetic field Ba (t) = Bdc + Bac cos(2π νt). Our calculated results
are compared with those of the critical state model (CSM) and found to prove the
approximate validity of the CSM below the irreversibility field. The differences between our
computed results and those of the CSM are also discussed.
1. Introduction
Since the discovery of high temperature superconductors
(HTSCs), most magnetic data have been taken on thin
films in the form of discs, slabs or strips with the applied
magnetic field perpendicular to the film surface. To obtain
maximum signal, most magnetization experiments are often
performed with thin specimens in a perpendicular field [2],
because in this orientation, even weak magnetic fields can
penetrate the film due to large demagnetization effects. One
may account for these geometry effects by introducing a
demagnetization factor. Sun et al [3] assumed the Bean
critical state model to describe the magnetic response of a thin
disc and obtained the hysteristic moment and temperature
dependent ac susceptibility.
An exact critical state model for thin discs in transverse
field was recently presented by Mikheenko and Kuzovlev [4].
They took into account the current distribution flowing in the
Meissner state (vortex-free region, Bz = 0) in addition to
the current density |J | = Jc , which is assumed constant
and flows in the vortex penetrated region and provided
analytical solutions for the magnetic response of the disc.
This model was extended by Zhu et al [5] who took into
account the fact that the current density should not change
abruptly from the vortex-penetrated region to the vortex-free
region, i.e.|J | Jc . Based on these two models, Clem
et al [1] gave a complete description of ac susceptibility
calculations for curves of χ ∼ Bac in the critical state model,
0953-2048/00/040356+06$30.00
© 2000 IOP Publishing Ltd
where Bac is the amplitude of the ac applied perpendicular
field. For ac magnetic response, Gurevich and Brandt [6]
pointed out that experimental results are usually analysed
within the framework of two opposite theoretical approaches.
The first one is based on the CSM below the irreversibility
line and the other is the linear flux dynamics above the
irreversibility line, where the current–voltage characteristic is
ohmic [7]. According to them, the CSM completely neglects
the dissipative processes in the critical or subcritical state
J Jc , where Jc (T , B) is assumed to be independent
of the electric field E. Thus the CSM can cause some
inaccuracy in describing the ac magnetic response of thin
superconducting films. This situation was changed after a
series of articles [8–10] were written by Brandt, in which
he developed a general formulation governing the behaviour
of thin superconducting films in axial field by deriving the
equation of motion for the sheet current using the E =
Ec (J /Jc )n material law equation with finite creep exponent
n. Most of the works by Brandt treated a finite thickness
of cylinder samples where the applied field is parallel to the
axis of the cylinder. Evidently, this can also be used in the
case of superconducting thin films. Hence, based on this
general formulation suggested by Brandt, it is possible to
deal with the problem of the extent to which the CSM is
approximately valid to analyse the experimental data for ac
magnetic responses.
In this paper, based on the general formulation by Brandt
for the sheet current in thin superconducting films, we attempt
Thin disc in axial field
to study the flux dynamics by numerical time integration
of the 1D integral equation of motion of the circulating
current density J(ρ, t) using the nonlinear E(J ) electric field
material law E = E(J )J/|J |.
In HTSCs, the CSM is only assumed to remain
approximately valid well below the irreversibility field.
Based on the CSM, [1] calculated the field and current
distributions, ac susceptibility components for n = 1, 3, 5
and the magnetization hysteresis loops as a function of
the ac field amplitude. Similarly, although temperaturedependent ac susceptibility responses are usually calculated,
we calculate the ac response in the flux creep regime as
a function of the ac field amplitude Bac for the purpose
of comparison and verification of the CSM below the
irreversibility line.
2. Theoretical model; equation of motion of the
sheet current
We consider a thin disc immersed in an applied timedependent magnetic field Ba (t) = Bdc + Bac cos(2π νt)
normal to the film surface, where Bac is the ac field amplitude
and Bdc is a constant dc biased field situated at the origin.
Evidently, shielding currents with current density J are
developed and flow in the plane of the disc in a circumferential
direction J = J eφ . The total vector potential due to current
and applied field as given by [2] is
A(ρ, t) =
R
0
G(ρ, ρ )J (ρ , t)d dρ +
ρ
Ba (t)
2
(1)
where d is the thickness of the film and G(ρ, ρ ) is the Green
function containing information about the specimen shape
and is defined as
ρ cos φ dφ µ0 π
G(ρ, ρ ) =
. (2)
2
2π 0 [ρ + ρ 2 − 2ρρ cos φ ]1/2
For the purpose of numerical calculations, we divide the disc
into N concentric loops of width R/N spaced equidistantly
and then approximate the average current density as J (i)
flowing in each loop with outer radius iR/N, inner radius (i−
1)R/N and average radius (i − 1/2)R/N [11]. Expressing
the induction law × E = −Ḃ = − ×Ȧ in the form
E = −Ȧ gives Ȧ = −E(J ), and the discretized form of
equation (1) becomes
−E(ρi , t) =
N
ρj
G(ρi , ρj )ρ J˙(ρj , t)d + Ḃa (t)
2
j =1
i = 1, 2, . . . , N.
G(ρi , ρj )G−1 (ρj , ρk )(ρ)2 = δik
where the prefactor Ec = BV0 and V0 is the flux creep
velocity. Next we introduce the current density J -dependent
activation energy
µ
Uc
Jc
U (J ) =
−1 .
(7)
µ
J
Equation (7) is a popular theoretical power law U (J )
relationship with much experimental support [12–14] that
describes the commonly used thermally activated processes,
where Uc is a characteristic energy scale and Jc is the critical
current density in the absence of thermal fluctuations. The
glassy exponent µ is a universal constant in the vortex-glass
theory with a value of about unity. In the collective-creep
theory, µ = 1/7 for the case of single vortex creep (J ∼ Jc );
µ = 3/2 for small bundle creep (J < Jc ) and µ = 7/9 in
the large bundle pinning regime (J Jc ) [15]. Besides,
equation (7) becomes U (J ) = Uc ln( JJc ) as µ → 0 and
U (J ) = Uc (1 − JJc ) for µ = −1 (Anderson–Kim model).
The electric field materials law therefore takes the form
µ
Uc
J
Jc
E = Ec
exp −
−1
(8)
Jc
T
J
where the extra factor J /Jc is introduced to provide a gradual
cross-over to the viscous flux flow regime [16, 17].
We can normalize equation (4) by introducing
, J˜ = JJc ,
dimensionless quantities: ρ̃ = Rρ , ρ̃ = ρ
R
Jc Rd
Ẽ = EEc , B̃ = BB0 and t˜ = tt0 where t0 = µ0 E
and
c
B0 = µ0 Jc d. Here, J˜ is in unit of Jc , B̃ in unit B0 = µ0 Jc d
RJc d
and t˜ in unit t0 = µ0 E
. Equation (4) can be therefore
c
rewritten as
N
ρ̃j ∂ B̃(t˜)
∂ J˜(ρ̃i , t˜) −1
˜
G̃ (ρ̃i , ρ̃j ) − Ẽ(ρ̃j , t ) −
=
2 ∂ t˜
∂ t˜
j =1
where
(3)
(4)
(5)
with ρ = R/N .
In the flux creep regime, the material law equation E(J )
becomes strongly nonlinear and can be generally described
by
U (J )
E(J ) = Ec exp −
.
(6)
T
i = 1, 2, . . . , N
In the general case of nonlinear E(J ) and arbitrary sweep
rate Ḃa (t), the time integration of equation (3) has to be
performed numerically, hence the time derivative should be
moved out from the integral to obtain J˙i (t) as an explicit
functional of J and Ḃa (t) [12]. Thus, inverting equation (3)
yields the following linear differential equation
N
ρj
˙ i , t)d =
J,(ρ
G−1 (ρi , ρj )ρ −E(ρj , t) − Ḃa (t)
2
j =1
i = 1, 2. . . . , N
where G−1 (ρi , ρj ) is related to G(ρ, ρ ) through the delta
function
G̃−1 (ρ̃i , ρ̃j ) = G−1 (ρi , ρj )µ0 Rρ.
(9)
(10)
Equation (9) completely describes the magnetic response
of thin superconducting films for some concrete cases of
Ba (t). For Ḃa (t) = 0, the last term vanishes and we obtain
the relaxation data if a critical state of Jc (ρ) = Jc (Ba ) =
constant is assumed for a biased Ba . If Ḃa (t) = constant and
the initial conditions are zero, then we can obtain the initial
magnetization of the sample, but in this case, the self-field of
the current density must be calculated too.
For the case of ac magnetic response, we have Ba (t) =
Bdc + Bac cos(2πνt), where ν is normalized as ν̃ = νν0 with
357
S A Aruna et al
c
ν0 = µ 0 E
. Substituting equation (8) into equation (9), we
RJc d
obtain
N
∂ J˜(ρ̃i , ν̃ t˜) G̃−1 (ρ̃i , ρ̃j )
=
∂(ν̃ t˜)
j =1
µ
˜
−J (ρ̃j , ν̃ t˜)
Uc
1
−1
exp −
×
ν̃
µT
J˜
ρ̃j
+ 2π B˜ac sin(2π ν̃ t˜)
(11)
2
Bdc = 0.5 T, the authors obtained the value of the parameter
µ = 0.62. Studies [18,19] have shown that at low fields
(Bdc = 0.5 T) in the vortex-glass-collective-creep model,
the Jc (T ) and Uc (T ) dependences are
Hereafter,
where G̃−1 (ρi , ρj ) = G−1 (ρi , ρj ) µ0NR .
neglecting the tildes over the dimensionless quantities in
equation (11), the equation of motion for the sheet current
becomes
N
∂J (ρi , νt) G−1 (ρi , ρj )
=
∂(νt)
j =1
µ
−J (ρj , νt)
Uc
1
−1
×
exp −
ν
µT
J
ρj
+ 2π Bac sin(2πνt)
i = 1, 2, . . . , N
(12)
2
which can be expressed in matrix form as
respectively, where Jc (0) = 9.06×1010 A m−2 , the activation
energy Uc (0) = 338 K, the flux creep velocity V0 = 10 m s−1
[18] and the reduced temperature t = T /Tc . In [18],
although experimental data for the parameters appearing in
the material law equation (7) are readily available for values
of Bdc in the range 0.5 to 6.0 T, we shall only make use of
the necessary data as given above at Bdc = 0.5 T for our
numerical calculations. For the other quantities, we chose
typical values for the disc radius R = 2.5×10−3 m, thickness
d = 2.0 × 10−7 m and N = 48.
Since we intend to verify the approximate validity of
the critical state model in calculating the ac susceptibility
dependence on ac magnetic field amplitudes Bac , we
chose an arbitrary fixed temperature T = 60 K well
below the irreversibility line, in which case thermally
activated flux creep can considerably affect the macroscopic
electrodynamics of these materials [7, 20] where the flux
creep is described by equation (7).
From equation (15), with Tc = 90 K and T = 60 K, we
obtained the temperature dependent critical current density
Jc (T = 60 K) = 1.73 × 1010 A m−2 and the corresponding
activation energy barrier Uc (T = 60 K) = 271.2 K.
Note that ν0 = µ0 RJEcc(T )d has different values for different
temperatures since Jc (T ) is temperature dependent, thus
in our case, we have ν0 (T = 60 K) = 4.59 × 105 Hz.
Equation (12) can then be readily integrated by the Runge–
Kutta method starting with Bac (t) = 0 and the known vector
Ji (0) at time t = 0, and then increasing Bac (t) gradually
to obtain the solution J (ρ, t). Note that this numerical
integration is stable if the time step dt is sufficiently small.
The magnetic moment m of the disc in dimensionless
quantities in transverse magnetic field has only a z
component, which can be calculated from
2
∂J (ρi , νt) −1
G (ρi , ρj )F (j )
=
∂(νt)
j
where F (j ) is given by
F (j ) =
−J (ρj , νt)
exp
ν
−
Uc
µT
(13)
µ
1
−1
J
ρj
(14)
2π Bac sin(2πνt).
2
We shall numerically time integrate this 1D integral equation
(equation (12)) for the sheet current to obtain the ac response
and then check the extent to which the critical state model is
valid below the irreversibility line.
+
3. Numerical calculations and discussion
We have performed computations of flux creep in a
uniform magnetic dc field Bdc , with an ac field Bac (t) =
Bac cos(2π νt) superimposed by numerical time integration
of equation (12). If the value of Bdc is large enough compared
with the self-field produced by shielding currents, then the
self-field and the ac field can be neglected and we can
approximate the field to be just equal to the dc field Bdc .
In order to calculate the ac susceptibility from equation (12),
the values of R and N must be known, and by combining
equations (2), (5) and (10), the quantity G̃(ρ̃i , ρ̃j ) can be
explicitly obtained. Meanwhile, the quantities µ, Uc /T , Jc ,
Bdc , Bac and ν appearing in equation (12) must be given too.
Also, to obtain the relationship between the practical and
normalized quantities Bac , ν, t, ρ, ρ and J , it is necessary to
know the values of the parameters in the expressions defining
the dimensionless quantities.
As an example, we shall make direct use of the data
from [18] for the above mentioned physical parameters in
equation (7), which have been obtained by the authors to fit
their experimental results for a YBa2 Cu3 O7 thin film whose
critical temperature Tc = 90 K. According to [18], for
358
Jc (T )
= (1 + t 2 )−1/2 (1 − t 2 )5/2
Jc (0)
(15)
Uc (T ) = Uc (0)[1 − t 4 ]
(16)
and
m=π
1
ρ 2 J (ρ) dρ.
(17)
0
From the periodic time-dependent magnetization Mz (t), the
ac susceptibility components χn and χn can be computed
from
2π
1
Mz (t) cos(2πnνt) d(2πνt) (18)
χn (Bac ) =
π Bac 0
and
χn (Bac ) =
1
πBac
2π
Mz (t) sin(2π nνt) d(2π νt).
(19)
0
Below, we shall present our typical calculation results
followed by discussions.
Thin disc in axial field
In the field sweeping process, the electric field E
induced at the sample perimeter is given in [22] as
0.4
χ''1/χ0
0.2
χ'1/χ0 & χ''1/χ0
0.0
χ'1/χ0
-0.2
-0.4
-0.6
Bdc=0.5 T
-0.8
f =5000 Hz
Peak: χ''1=0.3195 χ0
-1.0
-1.2
0.1
1
10
Bac
(a)
4
χ"3/χ0
χ'3/χ0
χ'3/χ0 & χ''3/χ0
2
0
-2
-4
-6
Bdc=0.5 T
-8
f= 5 kHz
-10
-12
0.01
0.1
1
10
Bac
(b)
0.010
χ'5/χ0
0.005
χ'5/χ''0 & χ''5/χ0
Shown in figure 1(a) are the normalized real (χ1 /χ0 )
and imaginary (χ1 /χ0 ) parts of the ac susceptibility versus
Bac at frequency f = 5000 Hz under Bdc = 0.5 T, where
χ0 is the real part of the ac susceptibility of fundamental
frequency response when Bac is small enough. The sharp
decrease in χ1 is a consequence of diamagnetic shielding
(inductive response), and the values of χ1 correspond to
energy dissipation in the sample (resistive response). The
peak value in χ1 is the point of maximum energy dissipation.
Our numerical calculations give this peak value χ1 =
0.3195χ0 , which is larger than that calculated by Clem et al
in [1] based on the CSM.
It is a well known fact that the nonlinear response of
the superconductor when flux pinning is present is reflected
in the presence of higher harmonics in the voltage wave
form and the appearance of odd susceptibility coefficients
(n = 3, 5, . . .). All the even harmonics are zero due to the
symmetry of the magnetization hysteresis loops. Figures 1(b)
and 1(c) show plots of the normalized real and imaginary parts
of the ac susceptibility components for third (n = 3) and fifth
(n = 5) harmonics respectively.
Further comparing our calculation results with those of
the CSM, we observed that the general features for the ac
susceptibility for n = 1, 3 and 5 are similar to those obtained
in the CSM. The differences are only quantitative in nature.
For instance, our results give the peak in χ1 at χ1 = 0.3195χ0
with a corresponding value of χ1 = −0.3977χ0 , while
χ1 = 0.241χ0 and χ1 = −0.382χ0 as calculated in [1]
based on the CSM. We also see that our calculation results
for χ3 and χ5 are very similar in appearance but with the
minimum in χ3 occurring at Bac < 1 while that in χ5
appears at Bac > 1. In contrast to our results, the minima
in both χ3 and χ5 in [1] appear at Bac > 1. Note that the
normalized value in [1] is Bd = µ0 Jc d/2, while in our case
it is B0 = µ0 Jc d, thus leading to a difference by a factor of
2 in comparing the corresponding values of Bac , which has
already been checked. Another major difference is the extra
wiggle appearing in χ5 near Bac = 1 in the CSM [1], which
is not evident in our result. Our calculation results give a
continuous curve at the frequency f = 5000 Hz.
Now we consider the frequency effect of ac
susceptibility. Figure 2 is a plot of (χ1 /χ0 ) and (χ1 /χ0 )
dependences on Bac at three frequencies f = 100 kHz,
5000 Hz, and 250 Hz under Bdc = 0.5 T. There are
two distinguished characters in these curves. First, both
the peak in χ1 and the step in χ1 shift to lower Bac with
decreasing frequency. Secondly, the width of the transition
in χ1 broadens as the frequency decreases in accordance with
experimental results. Our results are evidently beyond those
of the critical state model. However, Van der Beek and Kes
[21] pointed out that for nonlinear ac magnetic responses,
then one assumes that J is not Jc , but rather determined by
the actual form of U (J ), in other words, J is a function of
the field sweep rate J = J (Ḃ), which is expressed via the
thermally activated energy barrier U (J ) as follows
Ec
U [J (Ḃ)] = T ln
.
(20)
E
χ''5/χ0
0.000
-0.005
-0.010
Bdc=0.5T
-0.015
f =5 kHz
0.1
1
10
Bac
(c)
(χ1 )
and imaginary (χ1 ) parts of n = 1
(fundamental-frequency) complex ac susceptibility components
normalized to χ0 , as a function of ac field amplitude Bac = Bdc /B0
where B0 = µ0 Jc d at f = 5000 Hz and Bdc = 0.5 T. A peak
appears at χ1 = 0.3195χ0 with a corresponding χ1 = −0.3977χ0 .
(b) Real (χ3 ) and imaginary (χ3 ) parts of n = 3 (third harmonic)
complex susceptibility components normalized to χ0 , as a function
of ac field amplitude Bac as in (a) at f = 5000 Hz and
Bdc = 0.5 T. (c) Real (χ5 ) and imaginary (χ5 ) parts of n = 5
(fifth harmonic) complex susceptibility components normalized to
χ0 , as a function of the ac field amplitude Bac at f = 5000 Hz and
Bdc = 0.5 T.
Figure 1. (a) Real
359
S A Aruna et al
0.4
0.0
χ''1/χ0
0.2
-0.1
a
0.0
J/Jc
χ'1/χ0 & χ''1/χ0
-0.2
χ'1/χ0
-0.2
c
-0.4
b
b
c
-0.3
a
d
-0.6
Bdc= 0.5 T
a f =100 kHz
b f =5 kHz
c f =250 HZ
-0.8
-1.0
-0.5
-1.2
0.1
1
e
-0.4
-0.6
0.0
10
0.2
0.4
Bac
0.6
0.8
1.0
ρ /R
(a)
Figure 2. Normalized ac susceptibility components at
fundamental frequency (n = 1) under Bdc = 0.5 T at frequencies
(a) f = 100 kHz, (b) 5000 Hz, (c) 250 Hz.
0.6
0.4
0.6
J/Jc
0.2
0.4
f
e d
c
b
a
0.0
-0.2
0.2
g
Mz(t)/Msat
-0.4
0.0
-0.6
0.0
0.2
0.4
0.6
0.8
1.0
ρ /R
-0.2
a
b
(b)
c
d
e
-0.4
-0.6
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Figure 4. (a) Normalized azimuthal current density distributions
(J /Jc ) against radius ρ/R during one cycle in the disc in
increasing magnetic field, Bac (t): (a) 0.0499; (b) 0.2455;
(c) 0.4669; (d) 0.6426; and (e) 0.7943. (b) Normalized azimuthal
current density distributions (J /Jc ) against radius ρ/R during one
cycle in the disc in decreasing magnetic field, Bac (t): (a) 0.7943;
(b) 0.6425; (c) 0.2455; (d) 0.1329; (e) −0.2454; (f) −0.4669;
(g) −0.7943.
B'ac
Figure 3. Magnetization Mz (t) hysteresis loops normalized to the
saturation magnetization Msat = π/3 against the field
Bac
= Bac (t) − Bdc at different ac field amplitudes: (a) 0.1019;
(b) 0.2033; (c) 0.4019; (d) 0.5074; (e) 0.6342.
E(R) = (R/2)(dBac /dt). If we approximate E in the flux
penetration region as E(R), then we can actually determine
J (Ḃ) from equation (20). Our calculation results therefore
verify that the prediction of Van der Beek and Kes is a much
better approximation than the original Bean critical state
model, because in our case, the energy dissipation must be
taken into account since Uc (T )/T = 4.5 for T = 60 K.
This is a main conclusion of this paper, because according to
Brandt, if Uc /T (or n) is large enough (e.g. n = 51), then
the CSM can be approximated. A detailed discussion of this
problem is beyond the scope of this paper. Here, we only
emphasize that in order to compare our calculated results of
the frequency effect of ac susceptibility with those of the
CSM, it is necessary to use J (Ḃ) instead of Jc as suggested
by Van der Beek and Kes [21]. Detailed calculation results
360
of this problem and its comparison with the above model will
be considered elsewhere.
When an ac magnetic field is applied to the sample
surface, the sample is taken through a complete hysteresis
loop. Shown in figure 3 is a plot of the magnetization
hysteresis loops normalized to the saturation magnetization
Msat = π/3 versus Bac
= Ba (t) − Bdc at different field
amplitudes when the applied time-dependent field completes
one cycle. Here, the magnetization depends on Bac , thus
for small Bac values, the corresponding current distributions
are small. We clearly see that different ac field amplitudes
Bac imply different hysteresis loops. In other words, different
magnetizations give different J , hence from our calculations,
the magnetization Mz (t) depends on the field sweep rate.
This calculation verifies that the current J is actually field
sweep rate dependent, i.e. J = J (Ḃ) where Ḃ = 2π νBac .
Each current distribution corresponds to a different hysteresis
loop as shown in figure 3.
Figure 4(a) is a plot of the normalized current distribution
J /Jc versus ρ/R for increasing field Bac (t) values. The
weak applied magnetic field Bac (t) = 0.0499 simply induces
Thin disc in axial field
azimuthal screening supercurrents which flow in circular
patterns around the disc with virtually no flux penetration as
shown by curve a in figure 4(a). However, as the magnitude
of Bac (t) increases with time (Bac (t) = 0.7943), the current
penetrates deeper into the sample as shown by curve e in
figure 4(a). The corresponding current distribution for one
cycle in decreasing field is shown in figure 4(b), where
vortices penetrate in the opposite direction at Bac (t) =
−0.7943. Note that curve e in figure 4(a) is a mirror reflection
of the current distribution (f) in figure 4(b) for one cycle
with the same magnitude but opposite in sign. It is clearly
evident from this figure that J is smaller than Jc in the vortex
penetration region (see figure 4(b), curve f).
Since our calculated Uc /T is only about 4.5, this value is
not sufficiently large enough to satisfy the condition for using
the Bean critical state model. Therefore, our calculation
results simply show that completely neglecting the energy
dissipation in the Bean CSM cannot provide accurate results
for ac magnetic responses, especially at different ac field
frequencies.
4. Summary and conclusion
We have considered a thin superconducting disc in a uniform
perpendicular dc magnetic field Bdc superimposed with an
ac field component Bac cos 2π νt. We successfully invoked
the general formulation by Brandt for the equation of
motion for the azimuthal sheet current in a thin film disc,
and used the vortex-glass-collective-creep model U (J ) =
(Uc /µ)[(Jc /J )µ − 1] (where µ = 0.62) to numerically
time integrate the 1D equation of motion, from which we
obtained the current distribution, magnetization curves and
the ac susceptibility components (n = 1, 3 and 5) using
experimental data for YBa2 Cu3 O7 thin film [18]. The
differences between our calculated results and those of the
CSM [1] are only quantitative in nature.
We know that ac susceptibility response from the Bean
CSM is independent of frequency. If we consider the
collective-creep model, then from our calculations, the ac
response is actually frequency dependent and the Bean CSM
is not accurate in describing this property (i.e. flux-creep
model). From the arguments of Van der Beek and Kes, if
one uses J (Ḃ) instead of Jc in the flux penetration region,
the frequency-dependent ac response of thin films can still be
explained by this method even if the energy dissipations due
to thermally activated flux diffusion are considered.
Acknowledgments
This work was jointly supported by the National Centre for
Research and Development on Superconductivity of China
under grant No AJ-4012 and by the Chinese National Science
Foundation for Doctoral Education.
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