PHYSICAL REVIEW B
VOLUME 62, NUMBER 22
1 DECEMBER 2000-II
Spectral distribution of activation energies in YBa2Cu3O7À ␦ thin films
R. Warthmann, J. Albrecht, and H. Kronmüller
Max-Planck-Institut für Metallforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
Ch. Jooss
Institut für Materialphysik, Georg-August-Universität, Windausweg 2, D-37073 Göttingen, Germany
共Received 2 August 2000兲
Local magnetic relaxation experiments are carried out on YBa2Cu3O7⫺ ␦ thin films by magneto-optics. The
current density is calculated quantitatively by an inversion scheme of Biot Savart’s law which gives direct
information about the time evolution of the current density with a high spatial resolution of about 5 ␮m. We
determine the local activation energies U 0 (x,y) by fitting different relaxation laws at different positions 共x, y兲
in the film plane. This analysis yields a spectral distribution of the activation energies that is independent of the
applied model. Possible mechanisms which could lead to such a distribution are discussed.
In contrast to conventional low-T c superconductors thermal depinning of flux lines leads to a pronounced time decay
of the supercurrents in high-temperature superconductors at
finite temperatures. This process is described essentially by
two parameters: the critical current density j c up to which
the pinning force equals the Lorentz force and the activation
energy U depending on the pinning potential U 0 and on the
macroscopic current density that acts as a driving Lorentz
force on the flux lines. Global measurements, mostly done by
superconducting quantum interference device 共SQUID兲 magnetometers, often yield some deviations from a pure logarithmic time decay as proposed by Anderson and Kim.1 It is still
an open question, whether this is due to a more complex
U( j) law2,3 or to a spectral distribution of activation energies
as suggested by Hagen and Griessen4 and Theuss and
Kronmüller.5
In this paper we report on local magnetic relaxation experiments in YBa2Cu3O7⫺ ␦ 共YBCO兲 thin films by magnetooptics. In particular, quantitative results on the time evolution of the local current density with a high spatial resolution
of about 5 ␮m are given. This analysis has become possible
by a recently developed inversion scheme of Biot Savart’s
law which uses Fourier transformation and convolution theorem and is described in detail in Ref. 6. It allows us to calculate the local current density from the measured field distribution with a high accuracy of better than 5% and without
assuming any specific model for the critical state, e.g.,
Bean’s model.7 Compared to global SQUID measurements8
or to measurements by Hall sensor arrays9 with a spatial
resolution of about 50 ␮m in one dimension, much more
information about the flux-creep process can be gathered using quantitative magneto-optics. In this paper we give results
on the spatially resolved time evolution of the current density
and then extend our analysis to calculate the local pinning
potentials U 0 (x,y) from the observed time decay of the
current-density distribution. Though our data can be fitted by
different U( j) laws, we shall prove that a distribution of
activation energies exists.
The measurements presented in this paper were carried
out on a square-shaped YBCO thin film with thickness
d⫽300 nm and a side length of 3 mm. The film was evapo0163-1829/2000/62共22兲/15226共4兲/$15.00
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rated by a sputtering technique on a 10⫻10 mm2 SrTiO3
substrate and patterned afterwards by chemical etching.
As a sensitive layer for the visualization of the flux density distribution by magneto-optics10,11 an iron garnet film12
was used. In Fig. 1 the perpendicular component of the magnetic flux density is plotted as a grayscale image. The external field B ex⫽80 mT is applied after the sample was cooled
to T⫽5 K in zero field; this leads to flux penetration into
about half of the sample. The flux penetrates in a cushionlike
FIG. 1. Grayscale plot of the flux-density distribution in a
square-shaped YBCO thin film. The applied external field is
B ex ⫽80 mT and T⫽5 K. The grayscale from black to white corresponds to the value of the perpendicular magnetic flux ranging from
zero to the maximum. The full lines represent the current stream
lines. Also shown is a profile of the current density along the
dashed line. The sample is located between the two peaks, the finite
values of the current density outside are artifacts of the measurement technique.
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©2000 The American Physical Society
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SPECTRAL DISTRIBUTION OF ACTIVATION . . .
15 227
magnetometers, the flux front cannot penetrate further into
the sample and thus the current density is reduced in the
whole sample during the flux-creep process.
Determining the time evolution j(x,y,t) of the current
density with a high spatial resolution 共about 10 ␮m after
noise filtering兲 we are able to obtain the activation energies
U 0 (x,y) with the same spatial resolution. This is achieved by
fitting our data to the predicted decrease of the current density for different U( j) models. All models of flux creep assume the motion of flux lines to be thermally activated with
an Arrhenius-like relaxation rate
冉
冊
1 1
U共 j 兲
⫽ exp ⫺
.
␶ ␶0
k BT
FIG. 2. Grayscale plot of the changes in the absolute value of
the current-density distribution. This image was obtained by subtracting the initial current density distribution at t⫽2 s from the one
at t⫽300 s. Dark and bright parts therefore refer to areas where the
current density decreased or increased during flux creep, respectively.
form, which is typical for square-shaped samples. The discontinuity lines along the diagonals,13 where the current
stream lines bend sharply, are clearly visible. The roughly
constant critical current density can be clearly distinguished
from the shielding currents in the flux-free part of the
sample. In the upper right part of the sample the flux and
current distributions are altered due to a macroscopic defect.
This defect leads to two additional, nearly parabolic discontinuity lines, between which the current stream lines are
slightly curved inwards. For a detailed description of the
changes in the flux and current-density distribution caused
by macroscopic defects, see, e.g., Refs. 6, 13, and 14.
To measure the relaxation of the flux and current distribution by magneto-optics, a sequence of images was taken
for a constant time step up to a total time t⫽300 s. The first
image is acquired about 2 s after the field of B ex⫽80 mT was
applied to the zero-field-cooled sample. The temporal
changes in the flux density distribution can be obtained by
subtracting images measured for different times, as was done
before by Forkl.15 Furthermore, our quantitative determination of the supercurrents allows to visualize the changes in
the current-density distribution. Figure 2 shows this differential picture as a grayscale image. The area which was filled
up with magnetic flux directly after applying the external
field is depicted dark gray, corresponding to a reduction of
the local current density. However, the areas near the flux
front and along the discontinuity lines show an increase of
the current density 共the bright parts in Fig. 2兲. This increase
is due to the further penetration of the flux lines during the
flux-creep process and illustrates a special feature of the
partly penetrated state, namely, that the shielding currents
turn into critical currents. In the fully penetrated state, which
is mostly studied in relaxation measurements using SQUID
共1兲
Here, the characteristic relaxation rate ␶ 0 has to be considered as a ‘‘macroscopic’’ time scale rather than the microscopic attempt frequency.16 The models differ in the prediction of the U( j) law and therefore show a different form of
the decrease of the current density.
The earliest and most simple relaxation model was given
by Anderson and Kim,1 who proposed the activation energy
U to be a linear function of the current density
冉 冊
U 共 j 兲 ⫽U 0 1⫺
j
,
jc
共2兲
i.e., U⫽0 at j⫽ j c and U⫽U 0 at j⫽0. Equations 共1兲 and 共2兲
yields a logarithmic time decay of the current density,
冋
j 共 t 兲 ⫽ j c 1⫺
冉 冊册
k BT
t
ln
U0
t0
.
共3兲
The model of Kim and Anderson assumes the volume of the
moving flux line bundle to be constant in time. An improved
model should take into account that the bundle size increases
with decreasing current density. This was pointed out by
Beasley and co-workers.17 Zeldov et al.2 assumed a logarithmic U( j) law
U 共 j 兲 ⫽U 0 ln
冉冊
jc
,
j
共4兲
and obtained
冋
j 共 t 兲 ⫽ j c exp ⫺
冉 冊册
t
k BT
ln
U0
t0
共5兲
for the time decay of the current density. Feigel’man et al.
treated the flux creep in the framework of collective-pinning
theory3 and obtained an inverse power law
U 共 j 兲 ⫽U 0
冋冉 冊 册
jc
j
␮
⫺1
共6兲
in the case of single vortex pinning. The exponent ␮ depends
on the current density and according to Ref. 3 changes from
␮ ⫽1/7 to ␮ ⫽3/2 and ␮ ⫽7/9 during flux creep. However,
there is evidence that pinning in YBCO thin films may be
dominated by correlated pinning at planar defects18 and it is
therefore questionable whether the collective-pinning theory
is applicable in this case. The results given in this paper are
for ␮ ⫽7/9 which fits our data best. From Eqs. 共1兲 and 共6兲 the
time evolution of the current density is described by
WARTHMANN, ALBRECHT, KRONMÜLLER, AND JOOSS
15 228
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FIG. 3. Fit of the time decay of the current density for different
U( j) laws at a randomly chosen point of the sample. The values for
the fit parameters given in the plot are for the linear U( j) law. All
curves coincide within the linewidth.
冋
j 共 t 兲 ⫽ j c 1⫹
冉 冊册
t
k BT
ln
U0
t0
⫺1/␮
.
共7兲
The local activation energies are obtained by fitting either
Eqs. 共3兲, 共5兲, or 共7兲 for each data point 共x,y兲 to the measured
time evolution j(x,y,t) of the current density distribution.
Besides the activation energy U 0 (x,y), the critical current
density j 0 (x,y) and the characteristic time scale t 0 (x,y)
were used as fit parameters. The critical current density, i.e.,
j 0 (x,y,t⫽0), has to be treated as a fit parameter, because of
the time delay of t⫽2 s between applying the external field
and the acquisition of the first magneto-optical image.
Fits of our data by all three models are shown in Fig. 3,
where we plot the measured time decay of the current density
共points兲 and the fitted curves for the different U( j) laws at a
randomly chosen position 共x,y兲 on the sample. All three models show the same good agreement with the measured data.
The fit parameters which are given in Fig. 3 correspond to
the linear U( j) law. The logarithmic U( j) 共U 0
⫽19.86 meV, t 0 ⫽6.30 s, j 0 ⫽2.72⫻1011 A/m2兲 and the inverse power law 共U 0 ⫽23.07 meV, t 0 ⫽4.36 s, j 0 ⫽2.74
⫻1011 A/m2兲 yield similar values for the fit parameters.
The spatial distribution of the activation energy as obtained for the linear model of Kim and Anderson is depicted
in Fig. 4 as a grayscale image. Obviously, Eq. 共3兲 is only
valid in those parts of the sample where the superconductor
is in the critical state when relaxation starts. Therefore, the
values of U 0 obtained in the center of the sample 共bright
area兲 do not have any physical meaning and will be neglected in the further discussion. The different gray values in
the penetrated area show clearly the existence of a distribution of activation energies throughout the sample. The line
profile below the grayscale plot shows values in the range of
20–40 meV. An increase of the energy values towards the
flux front is visible.
The spectral distribution of the activation energies is plotted for the linear 共solid line兲, the logarithmic 共dashed line兲
and the inverse power 共dash-dotted line兲 U( j) law in Fig. 5.
The histograms were obtained by dividing the range of the
activation energies in N⫽100 equidistant intervals and
FIG. 4. Grayscale plot of the local activation energy U 0 (x,y).
Dark areas refer to low values and bright areas to high values of the
activation energy.
counting the number of data points in each channel. All distributions show some broadened peak with a maximum at
about 25 meV and a full width at half maximum of roughly
10 meV. Note, that the fit at each point in the sample was
done with a single value for the activation energy and the
spectral distribution arises from summing up over the whole
sample. Therefore, Fig. 5 clearly proves the existence of a
spectral distribution of activation energies.
We now discuss which mechanisms could lead to a distribution of the activation energies. First, one may consider a
field dependence of the activation energy. To investigate
FIG. 5. Distribution of activation energies for the linear 共solid
line兲, logarithmic 共dashed line兲 and inverse power law 共dash-dotted
line兲 U( j) law. The inset shows the magnetic-field dependence of
the activation energy obtained for the model of Anderson and Kim.
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SPECTRAL DISTRIBUTION OF ACTIVATION . . .
FIG. 6. Spectral distribution of activation energies at fixed local
flux density B z (x,y)⫽60 mT for the different U( j) laws. The solid
line represents the linear, the dashed line the logarithmic and the
dot-dashed line the inverse power law.
this, the activation energy is plotted versus the local magnetic field B z in the inset of Fig. 5. The curve is depicted for
the linear model of Kim and Anderson but similiar results
are obtained for the other models. A decrease of the activation energy with increasing local flux density is clearly
visible. U 0 drops from U 0 ⫽40 meV at B z ⫽15 mT to
U 0 ⫽20 meV at B z ⫽80 mT. This suggests that the activation
energy depends on the number of vortices, i.e., on the fluxbundle size moving collectively during flux creep.
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15 229
Finally, we consider the question, whether the distribution
of activation energies is additionally caused by variations of
the microstructure. For this analysis, one has to subtract the
field dependence 共inset of Fig. 5兲 from the distribution of
activation energies given in Fig. 5. This is done by determining the spectral distribution with the same method used in
Fig. 5 but for a fixed value of the local flux density. Figure 6
gives the distribution of the activation energies for all positions 共x, y兲 where B z (x,y)⫽60 mT. Note, that in the inset of
Fig. 5 the values of the activation energy are given by the
average values of distributions such as in Fig. 6. The existence of the similar distribution for all different U( j) laws
proves, that the distribution of activation energies is due to
both a field dependence and variations of the microstructure
of the sample.
In conclusion, we a presented quantitative method to determine the time evolution of the current density and the
activation energies in superconducting samples with high
spatial resolution. Our data can be described by different
models for flux creep. Independent of the U( j) law we obtained a similar spectral distribution of the activation energies. We discussed two mechanisms which can explain the
obtained distributions: 共i兲 variations in the microstructure
and 共ii兲 a dependence of U(x,y) on the local field B(x,y).
We showed that both mechanisms contribute to the distribution of the activation energies.
The authors are grateful to U. Sticher and H.-U. Habermeier for the preparation of the excellent YBCO thin films
and to E. H. Brandt for stimulating discussions.
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