Supercond. Sci. Technol. 12 (1999) 682–689. Printed in the UK
PII: S0953-2048(99)04672-2
Flux creep in a thin disc of
YBa2Cu3O7−δ superconductor
E Moraitakis, M Pissas, G Kallias and D Niarchos
Institute of Materials Science, National Centre for Scientific Research, ‘Demokritos’,
153 10 Aghia Paraskevi, Athens, Greece
Received 1 June 1999, in final form 23 July 1999
Abstract. We investigated the flux-creep behaviour of a thin disc of YBa2 Cu3 O7−δ
superconductor in the framework of an analytical model, in the temperature range
T = 10–60 K and for a perpendicular applied field Ha = 5–50 kG. The relaxation of the
magnetic moment shows a logarithmic dependence on time for the whole temperature range.
The field and temperature dependences of the critical current jc and the pinning potential U0
can be derived accurately by fitting the relaxation data. The results are discussed within
collective pinning theory. The critical current jc shows a power law field dependence of the
form jc (H ) ∝ H −b with b ' 0.50, which is kept during the relaxation process. The E versus
j characteristics are quantitatively derived from the relaxation data at the circumference of
the disc and show a very steep E(j ) relation described by a power law E(j ) = Ec (j/jc )n for
the whole temperature range, while a change of this behaviour is observed for T = 60 K. The
activation barrier U (j, T = 0, H ) is derived with the application of Maley’s method based on
the exact relations for a thin disc and this gives an exponent µ ' 0.68 for H = 20 kG.
1. Introduction
Thermal activation of vortices in high-Tc superconductors
(HTSs) yields to strong relaxation of the irreversible
magnetization limiting the current-carrying capacity of these
materials. The macroscopic electrodynamic properties of
HTS specimens become sensitive to the highly nonlinear
current–voltage characteristics (E versus j relation) in the
subcritical region (j < jc ), determined by microscopic
mechanisms of flux dynamics and pinning [1].
The case of a superconducting thin film in a
perpendicular magnetic field exhibits novel features as
compared with the well-known case of a slab in a parallel
field (parallel case) [2]. Static analytical calculations [3–10]
were carried out and numerical solutions [11–15] for the
current j (r ) and magnetic field B (r ) were also obtained for
a variety of sample geometries in a perpendicular applied
field. Flux dynamics in thin films in a perpendicular field has
attracted much interest and a detailed theoretical description
has been given for the linear E versus j relation (Ohmic
case) which holds in both the flux-flow regime and the TAFF
regime observed for low currents at temperatures above the
irreversibility line [16]. Also, a detailed analysis has been
given for the general nonlinear E versus j relation, well
below the irreversibility line [17, 18]. In the latter case, the
formulation developed in [17] permits a detailed description
of a thin strip or disc. It was shown that the flux diffusion is
one-dimensional, nonlinear and nonlocal and it is described
by an integral equation which can be solved analytically.
Exact relations for the relaxing electric field, for the relaxing
current density profiles and for the relaxing magnetic moment
are derived. Furthermore, the magnetic moment exhibits the
0953-2048/99/100682+08$30.00
© 1999 IOP Publishing Ltd
well-known logarithmic dependence on time for long-enough
times relative to characteristic ones. This formulation was
generalized in [18] in order to treat long superconductors of
arbitrary cross section with arbitrary E versus j relation. It
must be noted that most of the relaxation experiments have
made use of rectangular specimens. However, in this case
the patterns of the sheet current and of the flux density have
some peculiar features (especially at low fields) relative to
the one-dimensional distributions in circular discs and strips
that are essential in order to understand the flux dynamics in
superconductors [14].
In this paper magnetic relaxation measurements taken
for different magnetic fields in a thin disk of YBa2 Cu3 O7−δ
superconductor are presented and analysed in the framework
of [17]. The use of a specific (disc) geometry and an
analytical model which describes magnetic relaxation in
this case permits the accurate determination of the critical
current density jc and the pinning potential U0 . We also
derive the E versus j characteristics from the relaxation data
in a quantitative way and apply the method of Maley and
Willis [19] for the determination of the activation barrier
U (j, T = 0, H ), based on the exact relations for a thin disc.
2. Analysis of relaxation measurements
In the following we summarize the main topics from [17]
and derive analytical relations for the case of a disc in a
perpendicular field to be used for the analysis of our data.
For an HTS, thermal activation of vortices induces an electric
field which can be written in the form
E = Ec exp[−U (j )/kT ]
(1)
Flux creep in a thin disc of YBa2 Cu3 O7−δ
where U (j ) is a flux-creep potential barrier that vanishes at
j = jc and Ec is a crossover electric field between the fluxflow and flux-creep regimes which defines the critical current
jc by E(jc ) = Ec . The above relation determines the E
versus j characteristics in the subcritical region j < jc which
together with Maxwell equations completely determine the
electrodynamic properties of HTS specimens [8, 17, 18].
For Anderson–Kim flux creep [20] the potential barrier is
linearized as U (j ) = U0 (1−j/jc ) and leads to an exponential
E versus j relation: E = Ec exp[−U0 (1 − j/jc )/kT ].
For the vortex glass and collective creep models [1, 21, 22]
the potential barrier U (j ) is a highly nonlinear function of
j and is generally described by the so-called interpolation
formula U (j ) = (U0 /µ)[(jc /j )µ − 1], where the exponent
µ depends on the dimensionality and the particular fluxcreep regime [23]. In the case of a 3D collective creep
model the values µ = 1/7, 5/2 and 7/9 are proposed
for pinning of single vortices, small flux bundles and large
flux bundles respectively [1, 21]. In the case of vortex
glass model 0 < µ 6 1 holds [22]. For the case of a
superconducting thin disc under a perpendicular magnetic
field the formulation which was developed in [17] gives a onedimensional, nonlinear and nonlocal flux diffusion which is
described by the integral equation (in Gaussian units)
1 ∂(rE)
1 ∂Ha
=−
r ∂r Z c ∂t
2 a ∂J ∂E(u, t) E(k)
K(k)
− 2
−
du
(2)
c 0 ∂E
∂t
r −u
r +u
where E(k) and K(k) are complete elliptic integrals,
Rd
k = (4ru)1/2 /(r + u) and J (r) = 0 j (r, z) dz = j (r)d is
the sheet current density flowing in the disc plane.
However, since the ratio kT /U0 1 well below the irreversibility line, we can approximate ∂j/∂E
=j1 /E = (kT /U0 )jc /E over a wide region of E values except for exponentially small fields E < Eg ≈ Ec
exp[−U0 /(1 + µ)kT ] Ec . In this case, after some transient time τ , the relaxing electric field E(r , t) = f (r )g(t)
separates into a universal profile f (r ) and a time dependence g(t) = 1/(t + τ ). The relaxing current density
profile is then obtained by inserting this general electric
field into the material law j = j (E) and it is approximated by j (r, t) = jc − j1 ln[Ec /E(r, t)]. In this way the
relaxing magnetic
R a moment of the disc can be extracted:
m(t) = (π/c) 0 r 2 J (r, t) dr, where J (r, t) = j (r, t)d is
the sheet current in the disc. This formulation is exact for the Anderson–Kim linearized flux-creep barrier
U (j ) = U0 (1 − j/jc ) but holds equally well for a sufficiently
nonlinear E(j ) relation as it was described above. Only for
the Ohmic case (flux-flow regime) or the TAFF regime [24]
at j j1 does E(j ) become linear (E = ρj ), and the profiles of E and j attain a different, also universal form and
decay exponentially [16]. When the applied magnetic field
increases with constant ramp rate R = dHa /dt, the first term
in equation (2) dominates and induces a steady-state electric
field:
E(r) = −Rr/2c.
(3)
After the applied magnetic field is stabilized at t = 0, the first
term in equation (2) vanishes and the electric field E(r, t)
begins to decay with time, owing to the nonzero resistivity
of a superconductor for j < jc , caused by thermal activation
of vortices. The exact solution for the case of disc of radius
a is written in the form
E(r, t) =
2j1 ad
fdisc (η)
c2 (t + τ )
(4)
where fdisc (n) is a universal function defined in [17].
The transient period τ during which the relaxation of
E(r, t) depends on the initial condition equation (3) can be
determined by equating m(t) for t = 0 to the unrelaxed
(initial) magnetic moment
Z
π a
J (r, 0)r 2 dr
m(0) =
c 0
S
Ra
1 π a3
Jc 1 + S ln
(5)
−
=
c 3
2cEc
3
where Jc = jc d is the sheet current, S = kT /U0
and R = dHa /dt is the field ramp rate. The result is
τ = 2.568J1 /cR, where J1 = (kT /U0 )jc d. We finally
obtain
m(t) = m(0) − m1 ln(1 + t/τ )
(6)
where m(0) is the initial magnetic moment from equation (5)
and m1 = π a 3 jc dS/3c. For t τ (regime of steady-state
relaxation), the time τ in equation (6) cancels out and thus
m(t) becomes independent of the initial conditions, as can be
easily verified. The result is
m(t) = mc − m1 ln(t/t0 )
(7)
where t0 = 0.920jc daS/c2 Ec , mc = π a 3 jc d/3c and
m1 = π a 3 jc dS/3c. The above logarithmic expression is
exact for the whole time window for the Anderson–Kim flux
creep. The problem of magnetic relaxation in a thin disc,
in terms of the current–voltage characteristics and the initial
conditions, was also treated appropriately by Zhukov et al
[25, 26]. A similar approach was also applied in the case
of an infinite plate in a parallel field (longitudinal geometry)
[27]. A universal electric field profile E(r , t) was also found
in this case and the initial stage as well as the long-time stage
of the relaxation of the magnetization were discriminated and
observed experimentally.
Now both jc and t0 depend on the electric field criterion
Ec . Notice that instead of jc and Ec one can take
another pair jc0 and Ec0 related to the previous pair by
jc0 = jc − j1 ln(Ec /Ec0 ). This transformation does not change
equation (7), provided that the quantity mc + m1 ln t0 remains
constant [27]. In other words the above transformation links
two critical current densities jc and jc0 defined at different
electric field criteria. For t t0 one can write down
the interpolation formula for the whole process of magnetic
moment relaxation [1, 21]. This formulation clears out the
question about the origin of ‘microscopic’ times t0 , τ in the
relaxation process. Both of them depend on the material’s
parameters but τ depends also on the initial conditions R =
dHa /dt, through well-defined analytical relations for the case
of a disc.
683
E Moraitakis et al
3. Experimental details
High-quality thin films of YBCO superconductor were
deposited with a simple sputtering technique as reported
elsewhere [28]. The films were grown epitaxially on
LaAlO3 (100) substrates with the c-axis normal to the
substrate and exhibited very good superconducting properties
(Tc ∼ 89–92 K, 1Tc ∼ 1–2 K, Jc (77 K) > 1×106 A cm−2 ).
A thin YBCO disc with a diameter of 5 mm and a thickness of
400 nm was patterned using standard photolithographic–wet
etching techniques. No degradation of Tc ' 90.2 K,
1Tc ' 1.2 K, was measured after the patterning by
an ac susceptibility technique. Magnetic measurements
were performed with a commercial SQUID magnetometer
(Quantum Design MPMSII), applying a magnetic field
perpendicular to the film surface (parallel to the c-axis of
the film). A scan length of 2 cm was chosen in order to avoid
magnetic field inhomogeneities during the measurement
scan. In all the relaxation measurements the crystal was
cooled in zero field, and then the magnetic field was increased
to the value Ha with a ramp rate R = dHa /dt which depends
on the field value. Then the field was fixed (initial time t = 0)
and the relaxation of m(t) was measured within the time
window ti ' 102 s 6 t 6 tf ' 104 s. Hysteresis loops
were collected for the YBCO disc up to a maximum field of
50 kOe, with different sweep rates R = dHa /dt, the field
increment 1Ha being varied from 200 to 10 000 Oe.
4. Results and discussion
Figure 1 is a semilogarithmic plot of the variation of the
magnetic moment with time at T = 10 K, with applied
field Ha k c-axis and for the field range Ha = 5–50 kOe.
The main characteristic of the measurements is the linear
variation of m(t) in the semilogarithmic plot for all the
magnetic fields. Within experimental accuracy the same
logarithmic time dependence of m(t) is also observed for
the whole temperature range T = 10–60 K, examined in
this work. The data were fitted using equation (7) which
describes the steady-state relaxation. Fits are shown by
straight lines in figure 1. The following expression was used:
m(t) = −P1 + P2 ln(t P3 /P2 ), where P1 = −mc , P2 = −m1
are free parameters and P3 = cEc (πa 2 /2)/0.920 = 10.674
by fixing the electric field criterion at Ec = 1 µV cm−1 . Such
a procedure was also used in order to analyse the magnetic
relaxation measurements of an HgBa 2 CuO4+δ single crystal
[29]. Therefore the critical current jc and pinning potential
U0 are calculated from the fitting parameters. Different
values of Ec = 1–103 µV cm−1 change the estimated jc , U0
values by no more than 7% owing to the logarithmic relation
between Ec and jc . This means that the jc values are well
defined even if Ec was arbitrarily chosen. The estimated jc
values must be considered as extrapolated ones at t = t0 ,
assuming the same U (j ) relation, i.e. without any crossover
from one pinning regime to another. Furthermore, from the
estimated values jc = 1.534×107 A cm−2 and U0 = 625.2 K
for Ha = 10 kOe and R = 90.9 Oe s−1 , we estimate
τ ' 0.028 s and t0 ' 0.0023 s. Since the time for the
first measurement is ti ' 74 s τ , we cannot observe the
initial transient regime 0 < t < τ in our measurements,
684
described by equation (6). We observe only the steadystate relaxation regime described by equation (7), in which
the nonlinearity of U (j ) manifests itself in the logarithmic
relaxation of m(t). Furthermore, nonlogarithmic relaxation
of m(t) described by the so-called interpolation formula has
been observed in HTSs for very-long-time (of the order of
days) relaxation measurements [30]. In fact this deviation
from the logarithmic relaxation can be obtained in the above
model using the exact relation for ∂j/∂E in the integrand
of equation (2) [17]. Figure 2 shows the dependence of the
estimated critical current jc values on the applied magnetic
field at T = 10 K. In the same plot the current density j
calculated from the relaxed magnetic moment at t = 103 s is
shown. Both can be fitted better using a power law relation of
the form j (B) = aH −b with b ' 0.48–0.49, in the examined
field range from 5 to 50 kG. This means that the same
field dependence of the critical current jc is kept during the
relaxation process. Furthermore, the same field dependence
of the critical current with b ' 0.47–0.53 has been obtained
for different temperatures in the range T = 10–60 K. Figure 2
shows also the variation of the estimated U0 with the
applied magnetic field at T = 10 K. The estimated values
do not show any significant dependence on the magnetic
field in the field range from 5 to 50 kG. Based on the
above results the relaxation rate dm/d ln t = (kT /U0 )mc
as a function of the magnetic field decreases as the field
increases mainly as a result of the decrease of jc . A power
law with b ' 0.3 was also found by Griessen et al [31]
to describe the field dependence of the estimated critical
current jc up to B = 70 kOe in YBa2 Cu3 O7−δ thin films.
This much weaker field dependence than the exponential
decay jc ∝ exp(−B 3/2 ) or jc ∝ B −3 predicted by theory
[1, 21] in the small-bundle and large-bundle regimes was
considered as further evidence in favour of the singlevortex regime. However, one expects that the single-vortex
depinning current density does not depend on the magnetic
field.
Figure 3 shows the variation of the estimated jc values
with temperature. Full lines represent fits with an exponential
law jc (T ) = jc (0) exp(−T /T0 ), with about the same values
of T0 ' 25 and 24 K for H = 20 and 40 kG respectively. This
apparent linear variation of ln jc versus T was also observed
by other authors [32, 33] for the examined temperature range,
and it may be considered as a manifestation of collective
pinning and creep effects. It must be noted that quantum
creep effects which are important at low temperature do
not influence the data in the examined temperature range
T = 10–60 K, as pointed out in [31, 32]. According to
collective pinning theory, the effect of pinning by randomly
distributed weak pinning centres leads to a finite correlation
length Lc parallel to the external magnetic field which is
aligned in the c-axis direction, in our case. The appropriate
length is given by [1]
Lcc ∼
= εξab
j0
jc
1/2
(8)
where the anisotropy ratio is ε = (mab /mc )1/2 and
the
depairing current is j0 = c80 /12
√ Ginzburg–Landau
2
3π 2 λ2 ξ with λ2 = λ2L /2, ξ 2 = 0.54ξBCS
(clean limit)
Flux creep in a thin disc of YBa2 Cu3 O7−δ
50
-0.4 40
30
-0.5
20
m (emu)
-0.6
-0.7
-0.8
10
-0.9
-1.0
-1.1
5
10
2
10
3
10
4
t (sec)
Figure 1. Semilogarithmic plot of the relaxation of the magnetic
moment versus time for the YBCO thin disc at T = 10 K and for
perpendicular magnetic fields Ha = 5, 10, 20, 30, 40 and 50 kG.
Full lines are fits to the data as explained in the text.
800
2.0
600
1.0
U 0(K)
1.5
7
2
J c (10 A/cm )
700
500
0.5
400
0
10
20
30
40
50
H(kG)
Figure 2. Dependence of the critical current density on the
applied field at T = 10 K. The estimated critical current values are
represented by full squares and the relaxed ones at t = 1000 s by
open squares. In addition, the current density values calculated
from the hysteretic moment (Bean model) for sweep rate
R ≈ 20 Oe s−1 are shown by crosses. The same power law
dependence j (H ) ≈ H −b with b = 0.48–0.50 was found to fit the
data better, as shown by full curves. The estimated U0 (right-hand
axis) as a function of the applied magnetic field is shown by open
circles. The dotted curve is a guide for the eye. The error bars in
both cases are typically much less than the size of the centred
symbols.
the characteristic lengths at T = 0.
Taking the
anisotropy ratio ε ' 1/6, the corresponding coherence
length ξBCS = ξab (0) ' 15 Å, the penetration depth λL
= λab (0) ' 1400 Å and the estimated critical current from
the extrapolation of jc (T → 0) ≈ 1.1 × 107 A cm−2
for H = 40 kG, one obtains Lcc (T = 0) ≈ 21 Å. This
value is higher than the interlayer distance of CuO2 layers
d ' 12 Å and satisfies d < Lcc < εa0 ' 36 Å for
H = 40 kG, where a0 = (80 /B)1/2 is the measure of flux
line lattice cell constant, which indicates single-vortex line
pinning. This value is very close to Lcc ≈ 28 Å derived
by Wen et al [32] for YBCO thin films, while Thompson
et al [30] derived Lcc ≈ 14 Å for proton-irradiated YBCO
crystals and Civale et al [34] derived Lcc ≈ 40 Å for ‘clean’
YBCO crystals. Furthermore, the crossover field at T = 0
between single-vortex and small-bundle regimes is given by
[1] Bsb = βsb Hc2 (jc /j0 ), with βsb ≈ 5, and this gives a
value of Bsb ' 8.4 T for Hc2 ≈ 120 T in our case. As
the temperature increases thermal fluctuations of vortices
lead to a smoothing of the disorder potential and thereby
pinning will be reduced. This phenomenon is known as
thermal depinning, and it must be seen not as an abrupt
transition but rather as a continuous crossover from a pinned
to an unpinned situation. The extent of the single-vortex
pinning regime is limited by the single-vortex depinning
s
which is given by (weak pinning condition)
temperature Tdp
s
Tdp ≈ 0.7(jc /Gi j0 )1/2 Tc where Gi ≈ 10−2 is the Ginzburg
s
≈ 73 K using the
number for YBCO [1]. We estimated Tdp
values of jc and j0 defined above. For parameters typical for
YBCO, Blatter et al estimated that the single-vortex regime
is realized within the B–T plane for B < Bsb ≈ 6 T and
s
≈ 60 K [1]. Our relaxation measurements
for T < Tdp
fall in this field and temperature range, and the above results
may be in favour of this picture, although the possibility of a
vortex glass state for T < 60 K cannot be excluded. It should
also be noted that the examined field and temperature range is
below the irreversibility line which is defined primarily by the
deviation from reversible to irreversible magnetic behaviour.
Our measurements of the irreversibility line from the onset
of irreversibility in hysteresis loops for the YBCO disc give
a power law dependence Hirr ∝ (1 − T /Tc )3/2 which gives
an estimated value Hirr (T = 60 K) ' 6.6 T. The glass
transition describing a second-order transition from a vortex
glass to a vortex liquid phase is higher than the examined
field and temperature range. A scaling approach of E versus j
measurements according to the vortex glass model for YBCO
films gives a transition field BG (70 K) ' 5 T, as shown by
other authors [35, 36].
Figure 4 shows the variation of the estimated U0
values with temperature for fields H = 20 and 40 kG.
In the same plot the variation of the so-called normalized
relaxation rate S = d ln m/d ln t = (1/m) dm/d ln t,
calculated at t = 120 s, is also shown. The temperature
dependence of S steeply increases after a plateau. The
plateau values S = 0.025–0.030 fall in the ‘universal’ plateau
range observed and discussed by Malozemoff and Fisher
[23, 37]. The estimated pinning potential U0 shows a
maximum at the temperature where the normalized relaxation
rate S = d ln m/d ln t starts to increase steeply, which
shifts to lower temperatures for higher fields.
It is
expected that for temperatures higher than 60 K the pinning
potential rapidly decreases as the critical temperature Tc is
685
E Moraitakis et al
H =20 kG
H =40 kG
7
2
Jc(A/cm )
10
10
6
10
20
30
40
50
60
T(K)
Figure 3. Variation of the critical current density with
temperature. The estimated values are shown by full and open
squares for fields H = 20 and 40 kG respectively. Full lines
represent fits with an exponential law jc = jc (0) exp(−T /T0 )
with T0 ≈ 25 and 24 K respectively.
H=20 kG
2500
H=40 kG
U0(K)
2000
1500
1000
500
a
10
S=-(1/m)*(dm/dlnt)
0.07
20
30
40
50
60
30
40
50
60
H=20 kG
H=40 kG
0.06
0.05
0.04
0.03
0.02
b
10
20
T(K)
Figure 4. (a) The variation of the estimated U0 with temperature
for the YBCO thin disc. The estimated values are shown by full
and open squares for the fields H = 20 and 40 kG respectively.
The errors bars are typically much less than the size of the centred
symbols. (b) The variation of the normalized relaxation rate
S = −(1/m) dm/d ln t at t = 120 s.
reached. The same behaviour was also observed for melttextured YBCO samples [38]. Similar behaviour was also
observed by Wen et al [32] for a dynamical relaxation rate
Q = d ln js /d ln(dB/dt) in YBa2 Cu3 O7−δ and YBa2 Cu4 O8
thin films.
686
m(t) =
1 π a3
j (a, t)d
c 3
(9)
where j (a, t) = j (t) = jc d[1 − (kT /U0 ) ln(t/t0 )]
as deduced by direct comparison of equation (9) with
equation (7). The electric field at the circumference of the
disc for t τ is calculated using equation (4) for t τ and
dm/dt = −m1 /t as
E(a, t) =
3000
0.01
From magnetic relaxation measurements, E(j ) characteristics can be obtained for the disc geometry, using the
analysis of section 2. A sufficiently nonlinear E(j ) relation gives for t τ a nearly uniform current density profile
in the disc (except for a small region near the centre of the
disc) which can be approximated in most of the disc’s area
by a constant value, corresponding to the value j (a, t) at the
circumference of the disc [17, 18]. This value is correlated
with the hysteretic magnetic moment of the disc in the full
flux-penetrated state (Ha > Hp = 2πjc d/c) according to
6 fdisc (1) dm
cπ a 2 dt
(10)
where fdisc (1) can be approximated with fdisc (0.999 989)
≈0.832 046 [17]. In this way the E(j ) relation at the
circumference of the disc r = a can be derived. Magnetic
relaxation measurements in a well-defined geometry probe
the E(j ) characteristics at much lower electric field levels
(of the order of 10−13 V cm−1 ) than those experimentally
accessible in transport measurements. The results for
T = 10 K are shown in a double-logarithmic plot in figure 5.
The apparent linearity in the log E versus log j plots means
that the E versus j relation can be effectively described
by a power law dependence. In fact, it can be shown that
the interpolation formula for U (j ) yields for µ 1 a
logarithmic expression U (j ) ≈ U0 ln(jc /j ). In this case the
E versus j relation is E(j ) ≈ Ec (j/jc )n with n = U0 /kT
[8, 39]. This power law E(j ) relation may be considered as
a general one which describes different regimes from Ohmic
(n = 1) to Bean-like behaviour (n → ∞) [8, 18]. The full
lines in figure 5 represent power law fits using the above
relation. In the fitting procedure the jc values were fixed to
those obtained from the relaxation fits and Ec , n were the
free parameters. Successful fits were obtained with values
Ec = 5 × 10−7 –1 × 10−6 V cm−1 and n = 49–52 as shown
in the inset of figure 5. Such a very steep power law E(j )
relation must be considered actually as an exponential one
(given by the same expression as for Anderson–Kim flux
creep), which for the narrow interval of current j spanned
during the relaxation appears as a line in log E–log j scale.
This behaviour is consistent with the assumptions of the
theoretical model and the very high values U0 /kT ≈ 60
estimated from the relaxation data. The narrow interval of
current density j spanned during the magnetic relaxation
in our case does not permit us to test the E versus j
characteristics at higher current density j or electric field E
levels. Power law E versus j characteristics from magnetic
relaxation and field-sweep measurements, in YBCO discs
and rings, were also obtained by other authors [26, 40]. In
[41] E versus j curves extracted from magnetic relaxation
measurements on a YBCO ring have been combined with
direct I –V measurements to allow quantitative analysis of
Flux creep in a thin disc of YBa2 Cu3 O7−δ
10
-9
H=20 KOe
55
n 50
10
10
-11
10
-12
10
-12
10
-13
-13
4x10
20H(kG) 40
E (V/cm)
E (V/cm)
-11
45
0
10
10
-10
10
50 40 30 20
50K
45K 40K
60K
5
10
6
2
10
7
10K
10
7
2
Figure 5. Double-logarithmic plots of the E versus j
characteristics at the circumference of the disc, estimated from the
relaxation data at T = 10 K. The apparent linearity means that a
power law relation E(j ) = Ec (j/jc )n fits the data better, as shown
by full lines. The inset shows the variation of the exponent n with
field.
data spanning more than 10 decades in electric field. The data
were consistent with a current-density-dependent pinning
energy given from the interpolation formula with an exponent
µ ≈ 1/3 at T = 70 K. Figure 6 shows the estimated E(j )
characteristics at the circumference of the disc for different
temperatures and for a field H = 20 kG. Deviation from the
linear variation behaviour in the log E versus log j plots is
observed for T = 60 K. It must be taken into account that
this temperature is close to the single vortex–small bundle
s
boundary which is estimated to be Tdp
≈ 60 K [1].
As pointed out by Maley and Willis [19], the true shape
of the activation barrier for vortex motion U (j, T = 0, H )
can be determined from the relaxation data. In the following
we derive an equation for U (j, T = 0, H ) based on the
exact relations for a thin disc. The E versus j relation at
the circumference of the disc (r = a) can be written using
equation (1) as
(11)
Using equation (10) and equation (9) we obtain E(a, t)
=0.416(4ad/c2 ) dj (a, t)/dt. We finally obtain for a thin disc
U (j, T = 0, H )/k = −T [ln(dj/dt) − C]g −1 (T )
6
20K
J (A/cm )
J (A/cm )
U [j (a, t)]/k = −T [ln E(a, t) − ln Ec ].
30K
(12)
where C = ln(Ec c2 /1.664ad) and j ≡ j (a, t). The
crossover electric field between flux flow and creep can
be estimated from Ec ≈ (1/c)H ν0 X, where ν0 and
X are a characteristic attempt frequency and hopping
distance respectively. The function g(T ) contains the
separated temperature dependence of the activation energy
Figure 6. Double-logarithmic plots of the estimated E versus j
characteristics at the circumference of the disc for H = 20 kG and
for different temperatures in the range T = 10–60 K. Deviation
from linearity is observed for T = 60 K.
U (j, T , H ) = U (j, T = 0, H )g(T ). We tried the functions
g(T ) = 1 − (T /Tc )2 suggested by Tinkham [38, 42, 43] and
g(T ) = [1 − (T /Tc )2 ]1.5 used by others [44, 45]; the best
results were found for the second one for the determination
of U (j, T = 0, H ). First, one has to plot the second
part of equation (12) as a function of j for the relaxation
data at different temperatures and, secondly, the unknown
parameter C has to be determined by the condition that all
points have to lie on one smooth curve. Rough estimations
of the parameter C for different Ec values are C ' 19 and
26 for Ec = 1 and 1000 µV cm−1 respectively. Figure 7
shows the U (j, T = 0, H = 20 kG) relation determined
from the relaxation data at different temperatures by using
the function g(T ) = [1 − (T /Tc )2 ]1.5 ; a variation of the C
value changes slightly the obtained U (j ) curves as shown
in the inset of figure 7. The value C = 30 was found
to put the data better on the same curve with a slight
deviation of the data at T = 60 K. The full curve is a
fit of the multiple data sets with the interpolation formula
U (j ) = (U0 /µ)[(jc /j )µ − 1]. In the fitting procedure the
value jc (H = 20 kG, T → 0 K) ' 1.6 × 107 A cm−2 from
the extrapolation of the jc (T ) curve in figure 3 was fixed,
while U0 , µ were the free parameters. Our estimated values
U0 ' 342 K and µ ' 0.68 are similar to those reported
by Wen et al [32, 33] for YBCO thin films. However, the
value µ ' 0.68 deviates from the corresponding one for the
single-vortex regime µ = 1/7 and is intermediate between
single-vortex and small-bundle regime µ = 5/2 according to
collective pinning theory. One explanation of this deviation is
that the transition from single-vortex to small-bundle regimes
should be considered as a rather wide crossover instead of a
sharp boundary and so the application of Maley’s method
in our relaxation data gives an intermediate value for the
687
E Moraitakis et al
4
0.9
2.13 G/sec
60K
(U/k)/g(T)
3000
50K
2000
2
1.5
2
1.5
c=30,g(T)=[1-(T/Tc) )]
c=25,g(T)=[1-(T/Tc) )]
45K
2
c=30,g(T)=[1-(T/Tc) )]
40K
47.2 G/sec
2
H=20 kG
M (emu)
(U/k)/g(T)(K)
20.0 G/sec
1000
20
30
40
84.6 G/sec
0.3
1
10
100
dH a/dt (G/sec)
0
100
1000
2
6
2
g(T)=[1-(T/Tc) )]
1.5
2x10
9x10
30K
20K
0
6
6
j(A/cm )
1x10
H=20kG,c=30
6
4x10
15K
6
2
6x10
-2
10K
6
8x10
0
10
Figure 7. Plot of the U (j, T = 0, H = 20 kG) relation
determined from the relaxation data at different temperatures
according to Maley’s method for the case of a disc. The value
C = 30 was used. The inset is a double-logarithmic plot of the
U (j ) relation for different C values and different functions g(T )
as discussed in the text. The full curves are fits to the data with the
interpolation formula.
exponent µ. Although there is a slight deviation of data at
T = 60 K in figure 7, the relaxation of the magnetic moment
does not clearly show any change of the observed logarithmic
time dependence within the examined time window. This
discrepancy cannot be easily resolved.
Figure 8 shows hysteresis loops of the YBCO disc
at T = 10 K, collected for different field sweep rates
R = dHa /dt by varying the field increment. Sweep rates
in SQUID measurements are not actually constant but can
be approximated with some average values. It is obvious
that the m versus Ha curves for these sweep rates almost
coincide with each other. In fact the inset of figure 8
shows in a double-logarithmic plot the hysteretic moment
1m = m+ − m− = πa 3 j d/3c, calculated using the Bean
model, as a function of the sweep rate R. The hysteretic
moment 1m shows a rather weak dependence on R,
indicating that we scan a very steep part of the E(j ) curves
for these sweep rates. The current density values as a
function of the applied field or R = 20 Oe s−1 calculated
from the hysteretic moment (Bean model) are shown in
figure 2. A measurement of the magnetic moment with a
SQUID magnetometer when a hysteresis loop is formed lasts
approximately t ' 60 s after the applied field has stabilized
to the desired value. The corresponding current density value
is not the critical current jc but some relaxed value reached
during this time interval. The same power law dependence as
for the relaxation data, j (H ) ∝ H −b with the same exponent
b ' 0.5, was found to fit the data better. The power law
dependence of jc needs modification for low fields owing
to a singularity at H = 0. It should be noted that the
characteristic field above which the critical current jc flows in
most of the disc’s area and the magnetic moment is saturated
is Hc = 2πjc d/c. This field is typically no more than 600 G
for YBCO films at T = 5 K [9, 10]. This means that, for the
field range from 5 to 50 kG examined above, H Hc and the
disc should be considered in the full flux-penetrated state. As
shown in [10] the critical state of a thin disc, incorporating
20
30
40
50
H(kG)
j(A/cm )
688
10 kG
∆ m(emu)
4000
Figure 8. Hysteresis loops of the YBCO disc collected at
T = 10 K for different sweep rates R = dHa /dt by varying the
field increment. The inset shows the hysteretic moment 1m,
calculated using the Bean model as a function of the sweep rate R.
the dependence of jc on the applied field through a Kimlike expression jc (Ha ) = jc (0)/(1 + |Ha |/B0 ) and taking
into account self-field effects, successfully reproduces the
experimental hysteresis loops up to 15 kG for YBCO thin
films.
5. Conclusions
The flux-creep behaviour of a YBa2 Cu3 O7−δ thin disc for
the temperature range T = 10–60 K and for a perpendicular
applied field Ha = 5–50 kG was analysed in the framework of
the analytical model described in [17], as discussed in the text.
Within experimental accuracy, the relaxation of the magnetic
moment shows a logarithmic dependence on time as predicted
for a sufficiently steep E(j ) relation in the above model. The
critical current jc values show an exponential dependence
on temperature of the form jc (T ) = jc (0) exp(−T /T0 ) with
T0 ' 25 K, which may be considered as a manifestation
of collective pinning and creep effects. The temperature
dependence of the pinning potential U0 shows a maximum
which shifts to lower temperatures for higher fields and
may indicate a field-dependent crossover of the vortex state.
Furthermore, jc values show a power law dependence on field
of the form jc (H ) ∝ H −b with b ' 0.50, which holds during
the relaxation process. Using the above analysis the E versus
j characteristics for the disc geometry are quantitatively
obtained from the relaxation data, and show a very steep
E(j ) relation described by a power law E(j ) = Ec (j/jc )n
while a change in this behaviour is observed for T = 60 K.
Furthermore, the application of Maley’s method for the
extraction of the activation barrier U (j, T = 0, H ) based on
the exact relations for a thin disc gives an exponent µ ' 0.68
for H = 20 kG which lies between those for the single-vortex
(1/7) and the small-bundle (5/2) regimes.
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