Physica C 299 Ž1998. 215–230
High-resolution magneto-optical imaging of critical currents in
YBa 2 Cu 3 O 7yd thin films
Ch. Jooss ) , R. Warthmann, A. Forkl, H. Kronmuller
¨
Max-Planck-Institut fur
¨ Metallforschung, Heisenbergstr. 1, D-70569, Stuttgart, Germany
Received 7 October 1997; revised 1 December 1997; accepted 9 December 1997
Abstract
Magneto-optical investigation of flux penetration into type-II superconductors allows the determination of the local
critical current density jc by inversion of the Biot–Savart law. Due to the required computational effort, in the past, this
method was limited to low spatial resolution of the current density. In this paper, we present a fast inversion scheme using
Fast-Fourier-Transformation, which allows high spatial resolution imaging of current distributions. Due to the nonlocal
relation between magnetic field and current density, it is necessary to use a method that images field and current distribution
of superconductors as a whole, and enables high resolution at the same time. To demonstrate the power of our method, the
local current density in a YBa 2 Cu 3 O 7y d square and disk are imaged with high resolution and described in detail for
increasing and decreasing external magnetic fields. At low fields, the dependencies of jc Ž B . on the local magnetic field B
and of the average jc Žmagnetic moment. on the applied magnetic field show significant differences. Finally, we directly
image the local current density near macroscopic defects in a square and a disk. The observed current distributions near
defects significantly differ from the predictions of an extended Bean model. q 1998 Elsevier Science B.V.
PACS: 74.60.Jg; 78.20.Ls
Keywords: YBa 2 Cu 3 O 7y d thin films; Biot–Savart law; Magneto-optical imaging
1. Introduction
The critical current density in type-II superconductors is a fundamental parameter, which is not
only of interest for technical applications. It also
offers a wide field of basic physics, related to the
collective interaction of the vortex line lattice with
structural and thermal disorder. There are several
methods to determine jc which often do not agree in
their results. In general, one has to distinguish be-
)
Corresponding author.
tween the global critical current density jc, which is
measured by transport current or magnetization measurements, and presents some average over the whole
sample, and the local critical current density jc . This
can be determined by space-resolved measurements
like scanning Hall probes w1,2x or magneto-optics
w3–5x. The differences between both critical current
densities are caused by inhomogeneities in the superconductor, like macroscopic defects, weak links,
scratches and by the local-field dependence of jc Ž B ..
At external fields Hex ) Hc1 , quantized magnetic
flux penetrates into a type-II superconductor. Due to
the pinning forces acting on the flux lines, flux
0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 9 2 1 - 4 5 3 4 Ž 9 7 . 0 1 8 8 7 - X
Ch. Jooss et al.r Physica C 299 (1998) 215–230
216
cannot fill up the sample homogeneously. Therefore,
a flux front builds up, penetrating the sample up to a
certain penetration depth determined by the balance
of pinning and driving forces f p q f d s 0 on the flux
lines. In high-k materials, the driving force density
f d is a Lorentz force density w6x of a macroscopic
current, acting on the magnetic induction of the
vortex lines. This macroscopic current is caused by
an inhomogeneous distribution of tiny superconducting eddy currents, and is thus related to gradients in
the vortex line density. In equilibrium state Žwhich is
truly a metastable state, relaxing due to thermal
activated flux creep., this macroscopic current density is the critical current density determined by the
pinning forces, hence f d s jc = B s yf p .
The macroscopic current distribution causes a
magnetic self-field of the sample according to Biot–
Savart’s law
1
j Ž rX . = Ž r y rX . 3 X
Hself s
d r.
Ž 1.
4p V
< r y rX < 3
The local magnetic flux density B Ž r . is a superposition of the external magnetic field m 0 Hex and the
self-field of the currents
B Ž r . s m 0 Hself q m 0 Hex .
Ž 2.
Hence, in principle the local critical current density can be determined by measuring the self-field,
when an external field is applied to the superconductor. Due to its high spatial resolution up to 1 m m, the
magneto-optical Faraday effect is an excellent tool
for investigating the self-field of a superconductor.
In this way, the local critical current density w7x,
two-dimensional current distributions w8x, and
anisotropies in the critical current density w9–11x can
be visualized. By using suitable calibration methods
of the measured light intensity w7x, the measurement
of the z-component of the magnetic induction Ž z
denotes the coordinate perpendicular to the film
plane. is possible with sufficient accuracy in a plane
z s const close to the film surface.
In general, there are two different methods to
extract jc from the measured field distribution. The
first one is to fit current distributions by comparing
calculated field distributions Bz Ž r . with measured
ones w8,12–14x. The second general method is based
on inversion of Biot–Savart’s law and yields directly
the current distribution from the measured Bz-distribution.
H
This inversion is a nontrivial problem, since in
thin films the critical current density according to
Ampere’s
law m 0 j s rot B is determined by the in`
plane gradients EB xrE z and EB yrE z w13x, respectively. The in-plane components B x and B y usually
cannot be measured by means of magneto-optical
methods; therefore, the local relation m 0 j s rot B
cannot be used to determine the current density.
However, it is possible to derive an integral relation
between the measured Bz Ž x, y .-data and the current
density j Ž x, y . in the film plane, if the z-dependence
of the current density across the film can be neglected. In this case, the Biot–Savart’s relation between the z-component of the magnetic induction Bz
and the current density j Ž x, y . in the film plane can
be inverted.
This paper is organized as follows: First we describe previously used inversion methods of Biot–
Savart’s law like matrix inversion w2,15–18x and
Fast-Fourier-Transformation ŽFFT. for infinitely thin
films w19x. Then, we present our own method, which
is an extension of the method of Roth et al. w20x to
films with finite thickness. We demonstrate the feasibility of the high-resolution inversion for a detailed
study of the local current in a YBaCuO square and
disk both containing inhomogeneities. It is already
known that there is a screening current in the flux
free area of thin films in addition to a critical current
due to vortex pinning for flux penetration in the
ZFC-case. We show now that this screening current
is also present during flux reversal, as theoretically
predicted by Brandt and Indenbom w21x and Zhu et
al. w22x, although the full sample is in the Shubnikov
state. Secondly, the local-field dependence of the
local current density jc Ž Bloc . is investigated. For low
fields, this significantly differs from the global-field
dependence of jcŽ Bex . usually measured by means of
a magnetometer. Finally, we directly image the true
local currents near macroscopic defects.
2. Inversion of Biot–Savart’s law for thin films
In thin superconducting films with film thickness
d smaller than the magnetic penetration depth perpendicular to the film Ž x, y .-plane, the current flows
in the film plane j Ž x, y, z . s e x j x Ž x, y . q e y j y Ž x, y .
and the z-dependence of the current density can be
Ch. Jooss et al.r Physica C 299 (1998) 215–230
neglected. Since we regard isolated superconducting
samples without circuit lines for a transport current,
the current stream lines are closed within the area of
the magnetic field measurement. In this case, one has
div j s 0 all over the measurement area and thus
may derive the current density from a scalar function
g Ž x, y . according to
j Ž x , y . s = = zg
ˆ Ž x, y. .
Ž 3.
Because of freedom of gauge choice, g in general is
defined only apart from a gradient term and an
integration constant, since the transformation g ™ g
q Ez c q c leaves j Ž x, y . invariant. The Coulomb
gauge condition requires Ez g s 0, which is fulfilled
by the ansatz Ž3., because g Ž x, y . has no z-dependence. Hence, our local magnetization g is unequivocally defined and g is a potential function for the
current density.
The magnetic moment of the sample is given by
1
ms
r = j Ž r . d 3 r s zˆ g Ž x , y . d 3 r .
Ž 4.
2 V
V
The relation between the scalar function g Ž x, y . and
the z-component of the local induction Bz Ž r . measured by means of magneto-optics is
H
H
Bz Ž r . s m 0 Hex q m 0 K g Ž r ,rX . g Ž xX , yX . d 3 r X , Ž 5 .
HV
where K g Ž r,rX . is an integral kernel,
K g Ž r ,rX .
1
s
2
2
2 Ž z y z X . y Ž x y x X . y Ž y y yX .
4p Ž x y x X . 2 q Ž y y y X . 2 q Ž z y z X . 2
2
5r2
.
Ž 6.
Eq. Ž5. can be formally inverted to give
1
g Ž x, y. s
Ky1 Ž r ,rX .
m0 V g
H
= Bz Ž xX , yX . y m 0 Hex d 3 r X .
Ž 7.
Here Ky1
is the inverted integral kernel, which may
g
be computed by different numerical methods as described in Sections 2.1, 2.2 and 2.3.
2.1. PreÕiously used methods
The measurement of the magnetic flux density
distribution yields an array of discrete Bz data points
217
on a grid. Therefore all methods of inversion of
Biot–Savart’s law introduce a grid in the Ž x, y .-plane
of the superconductor and result in a matrix equation
for Eq. Ž7. in r-space or in a discrete Fourier-transformation of a scalar function on a grid in k-space.
For the calculation of g Ž x, y . we define an
equidistant grid x i s Ž i y 1r2.WxrNx with i s
1 . . . Nx and yj s Ž j y 1r2.WyrNy with j s 1 . . . Ny .
Using a grid the sample is divided into N s Nx = Ny
square cells, each covering an area p = p in the
Ž x, y .-plane with pixel-size p s WxrNx and thickness d. Within each cell, the Bz Ž i, j .- or g Ž i, j .-value
is constant. Therefore, g Ž i, j . in Eq. Ž7. can be
written as
gi j s
1
m0
mn
Ý Qy1
i jm n Ž B z y m 0 Hex .
Ž 8.
m, n
where the four-dimensional matrix Q is symmetric
and nonsingular, but not positively definite.
The calculation of currents by matrix inversion
was first achieved by Grant et al. w15x, assuming that
the current is flowing in an infinitely thin layer. They
obtained the current stream lines and the critical
current density in square and rectangular TBCCO
and YBaCuO thin films. The field distribution
Bz Ž k,l . was measured by a standard Hall probe on a
grid of 41 = 41 datapoints. The current distribution
of polycrystalline high-Tc materials was investigated
by inversion method by Niculescu et al. w2x. They
could determine quantitatively the distribution of the
critical current density with a resolution of 1 mm2 ,
using a diagonal approximation of the matrix Q. This
results in a local relation between the field and the
current density, and is a very rough approximation.
Furthermore, Brandt w16,17x developed an inversion
scheme for an infinitely thin film by using a Fourier
series for QŽ r,rX ..
All described inversion methods above have the
principal problem that the spatial resolution of the
field data Bz Ž k,l ., given on an Nx = Ny grid, is
limited to relatively small sizes of Nx s Ny s 50 to
80, since calculation and storage expense should not
be too large when using conventional workstations.
As a result of the Nx2 Ny2 scaling of the matrix size of
Q, the inversion of Bz Ž k,l . data of 100 = 100 datapoints would require 400 Mbyte storage for Q calculated in 4 byte floating numbers.
Ch. Jooss et al.r Physica C 299 (1998) 215–230
218
The first progress in developing a more efficient
inversion scheme was reported by Wijngaarden et al.
w18x. Using the translational invariance of Biot–
Savart’s law, they detected that Q is a four-dimensional Toeplitz matrix. The symmetry is useful for
efficient storage of Q, which then scales with Nx Ny
instead of Nx2 Ny2 , but they did not use this symmetry
for the calculation of Qy1 . Wijngaarden et al. observe an increase of CPU time for computation with
Na4.5 Ž a s x or y . using a standard conjugate gradient method and a linear speedup with the number of
processors.
The inversion problem is solved most efficiently
by using the translational invariance of Biot–Savart’s
law. This allows the application of convolution theorem in k-space. The method was first described by
Roth et al. w20x but was not applied to superconductors. Pashitski et al. w19x used the method of Roth for
the investigation of the current distribution near a
grain boundary in a TlBa 2 Ca 2 Cu 3 O x thin film. Like
Brandt w17x and Grant w15x, they used a sheet current
J s dj and the integral kernel of Biot–Savart’s law
for infinitely thin films. However, our tests of this
method show that the finite thickness of the samples
has to be considered to get quantitatively correct
results. For an infinitely thin film, logarithmic singularities in Bz occur at the sample edges. In contrast,
the measured Bz data for films with finite thickness
show finite rounded peaks at the sample edges.
Thus, to obtain correct current densities at the edges,
the inversion method must take into account the
finite sample thickness, and one has to use a different integral kernel.
2.2. Fast inÕersion using the conÕolution theorem
The translational symmetry of the kernel K g in
Eq. Ž5. of Biot–Savart’s law allows to use the
convolution theorem for the Fourier transformed kernels. If one writes K g Ž r,rX . s K g Ž r y rX . and extend
the integration volume to infinity in Eq. Ž5., one
yields in r- and k-space, respectively,
Bz Ž r . y m 0 Hex s m 0
HV™` K
B˜z Ž k . s m o K˜ g Ž k . g˜ Ž k . .
g
Ž r y rX . g Ž rX . d 3 r X ,
therefore the inversion problem is reduced to the
calculation of Nx Ny components of the kernels in
k-space instead of Nx2 Ny2 for matrix inversion. Since
we assume that the current density is not varying in
z-direction inside the film, it is possible to use
two-dimensional Fourier transformations
`
B̃z Ž k x ,k y . s
`
Hy`d xHy`d yB Ž x , y . e
z
`
g˜ Ž k x ,k y . s
`
Hy`d xHy`d yg Ž x , y . e
iŽ k x xqk y y.
iŽ k x xqk y y.
,
,
Ž 10 .
while the integration over the film thickness d remains. Then Eq. Ž5. can be transformed into
m 0 dr2 X ykŽ hyzX .
B̃z Ž k x ,k y ,h . s
d z ke
2 ydr2
H
=g˜ Ž k x ,k y , zX . ,
Ž 11 .
using the convolution theorem for the Ž x, y .-plane of
the thin film and k s Ž k 2x q k 2y .1r2 . The measurement
height h of the field distribution has to be taken into
account. Assuming a g˜ not depending on zX , one can
integrate Eq. Ž11. easily, and the problem of the
logarithmic edge singularities of the infinitely thin
film is eliminated. We obtain
B˜z Ž k x ,k y ,h . s m 0 K˜ g Ž k x ,k y ,h,d . g˜ Ž k x ,k y . ,
Ž 12 .
with Fourier-transformed integral kernel
kd
K̃ g s eyk h sinh
.
Ž 13 .
2
The inverse relation between g˜ and B˜z is simply
given by
ž /
g˜ s
B̃z
m 0 K˜ g
.
Ž 14 .
Alternatively, the inversion can also be performed
directly for the two current density components j˜x
and j˜y w20x. Since the finite thickness is taken into
account, one obtains
j˜x s yi
B̃z
K˜ x
,
j˜y s yj˜x
kx
ky
.
Ž 15 .
with
Ž 9.
In k-space, the integrals over the j or g distributions reduce to a simple product of two functions;
K̃ x Ž k x ,k y ,h,d . s m 0
eyk h
k
sinh
kd
ky
2
k
ž /
q
k 2x
kyk
Ž 16 .
Ch. Jooss et al.r Physica C 299 (1998) 215–230
and using the continuity equation k x j˜x Ž k x ,k y .
qk y j˜y Ž k x ,k y . s 0 in k-space. In this picture, the
current density is just a high-pass-filtered version of
the magnetic field distribution, and the integral kernels in k-space can be considered as filter functions.
The calculation expense of this exact inversion
scheme consists of two FFTs and filtering, which
results in a Nx Ny Ž1 q 2logŽ Nx Ny .. scaling, using
fast Fourier transform algorithms instead of Ž Nx Ny . 3
for matrix inversion or Ž Nx Ny . 2.25 using the Toeplitz
symmetry.
The application of Fourier-transformation and
convolution theorem requires a periodic continuation
of the obtained current distributions, since one is not
able to measure the field distribution in the infinite
plane. Therefore, the restriction of the calculation of
the Fourier transformed quantities to a finite area
with Nx Ny data points instead of using the exact Eq.
Ž10. introduces a superlattice that consists of a periodic array of identical current distributions. To avoid
a significant interaction between the supercells, the
current-carrying sample must be smaller than the
measured field image. Therefore, the magneto-optical field measurement should cover an area larger
than the sample. The artefacts introduced by the
219
supercell are very small if the ratio of the sample
size to the area of the measured field data is f 0.5
or smaller.
The presence of noise in the measured data gives
rise to additional complications during the inversion.
The frequency spectrum of the measured magnetic
field is dominated by noise at high wave numbers,
and is strongly amplified by the high-pass filter K˜y1
g .
Without low-pass-filtering, the measured magnetic
field data, the current distribution would be dominated by noise. In Ref. w20x, the magnetic field in
k-space was multiplied by a Hanning window to
reduce the high-frequency noise of the input data,
WŽ k.
°
¢
0.5 1 q cos
s~
0
pk
ž /
k max
for
k - k max ,
for
k ) k max .
Ž 17 .
The cutoff wave number k max has to be determined
empirically and reduces the resolution of the calculated current distribution. An increased measurement
height above the specimen amplifies the high-pass
Fig. 1. Greyscale image of the B z distribution inside and outside a square thin film of width 2 mm, d s 200 nm and h s 500 nm on a grid
of 1400 = 1400 points. The current stream lines, obtained by inversion with the convolution theorem, are shown as black lines. Profiles of
the two obtained current components along the two crossed lines are plotted below Ž j y . and right Ž j x ..
220
Ch. Jooss et al.r Physica C 299 (1998) 215–230
function of K˜y1 and thus requires a stronger lowpass-filtering of the field data. Thus, noise and measurement height limit the spatial resolution, which
can be achieved by the inversion.
2.3. Test of the inÕersion scheme
To check the accuracy of our inversion method,
the field distribution of a known current distribution
in a square film was calculated using the method of
basic current-bricks published by Forkl w8x. In Fig. 1,
the flux density distribution for the fully penetrated
state of a square with jc s const.s 3.5 = 10 11 Arm2
is shown for a spatial resolution of 1400 = 1400
datapoints. The plane above the square, where the
flux density is calculated is given by z s h s 500
nm. In our simulation, the sample has a width of
Wx s Wy s 2 mm and a thickness of 200 nm. The
current stream lines and the current density, which is
plotted along two lines, is obtained by Fourier-transformation on a Nx s Ny s 1400 grid. Each pixel has
a size p s 2.86 m m. After application of the Hanning window, the spatial resolution is 2p ky1
max s 4
m m.
The obtained current distribution is in excellent
agreement with the input current. The current density
at the current plateau is 3.5 " 0.03 P 10 11 Arm2 all
over the sample. There are some small distortions in
the magnitude of the current of 2–3 P 10 9 due to
noise. These distortions are two orders of magnitude
smaller than the input current density and could be
reduced much more by means of a hanning window.
The square covers only a part of 700 = 700 pixels of
the grid and is located at the center of the image. In
this case, the influence of the supercell interaction on
the obtained current density is negligibly small. For
the inversion of experimental data, the ratio of sample size to measurement area can be improved by an
artificial enlargement of the image size outside the
sample by linear extrapolation of the self-field of the
sample to zero.
3. Field detection and current imaging
3.1. Magneto-optical field detection
As field detecting elements, we use two different
magneto-optical layers ŽMOLs.: A ferrimagnetic
iron–garnet film with in-plane anisotropy w23x or
EuSe w4x, which is evaporated onto the superconductor and allows a high spatial resolution of about 1
m m. Before evaporating EuSe on YBaCuO, a gold
layer of 200 nm has to be sputtered on the superconductor, because YBaCuO is strongly absorbing at the
visible light wavelengths. Therefore, using EuSe, the
measurement height h is between 200 nm and 450
nm above the surface of the superconductor. The
iron–garnet film exhibits a higher sensitivity for
field detection, but has the disadvantage of a lower
spatial resolution than EuSe. Furthermore, the measurement height is not exactly known. A quantitative
analysis based on the comparison of field measurements using EuSe with field measurements by the
garnet indicator, allows an estimate of the measurement height of h s 5–10 m m. Due to this uncertainty, the critical current density can be determined
with an accuracy of only 5%. Before calibrating the
light intensities, a background image of the ZFC-state
is subtracted from the magneto-optical image, to
eliminate all non-field-depending distortions. The
calibration functions for MOLs, EuSe and garnet
indicator are described in Ref. w7x. The remaining
uncertainties of the field-calibration have a magnitude of 5% to 10%, and are caused by the field-dependent Faraday rotation in the objective lenses,
polarizers in the microscope, and in the glass, covering the cryostat.
3.2. Current-imaging in YBaCuO thin films
We present in this section current distributions in
a YBaCuO thin disk with a diameter of 2 mm and a
film thickness of 290 nm. The measurements of the
Bz-distribution are performed with EuSe as MOL at
He-temperature Ž4.2 K.. The external field is applied
perpendicular to the film plane and is increased from
the ZFC-state Ž Bex s 0. to a maximum field Bex,max .
Afterwards it is successively decreased until approaching the remanent state Ž Bex s 0..
Fig. 2 shows a superposition of the measured
magnetic flux density distribution as a greyscale
image and the current stream lines of the superconducting current flowing in the sample for different
external fields. Additionally, profiles of the angular
component of the current density jf along the
dashed–dotted line are plotted below each image. In
Ch. Jooss et al.r Physica C 299 (1998) 215–230
order to reduce noise, which dominates the highfrequency spectrum of the measured magnetic flux
density in k-space, the B˜z data is filtered by means
of the Hanning window, resulting in the smallest
resolved wavelength of the obtained current distribution of l min s 6.1 m m. Outside the disk, the inver-
221
sion yields also an apparent non-zero current density
of less than 1 = 10 10 Arm2 , which is at least one
order of magnitude smaller than the current density
inside the specimen. This artefact is mainly caused
by a different light reflection at the Au layer covering the YBaCuO film, and the SrTiO 3 substrate
Fig. 2. Greyscale representation of the measured B z distributions of a YBaCuO disk for Ža. Be x s 16.0 mT, Žb. 48.8 mT, Žc. 176 mT and
Žd. 47.2 mT. The superimposed computed current stream lines are visible as white or black lines. The dark spot f 400 m m right from the
disk center in Žc. and Žd. mark a distortion in the magneto-optical layer, which influences the obtained current density locally. Also shown
are plots of the jf- profiles along the dot–dashed dark line.
222
Ch. Jooss et al.r Physica C 299 (1998) 215–230
outside the sample. However, the consequences for
the computed current distribution near the disk borders are negligible, because of the smallness of this
effect.
Usually, in ideal superconducting disks, the current is described by means of a one-dimensional
current density jf Ž r . w12,22,24–27x. The application
of the inversion of Biot–Savart’s law enables us now
to derive the two-dimensional current density Ž jr , jf .
in the disk. Therefore, deviations from the one-dimensional model and the occurrence of a radial
current component can be determined. These deviations are clearly visible in the current stream lines
plotted in Figs. 2 and 3. In the latter, the jf- and
jr-components are visualized as greyscale images for
a state, where flux has partly penetrated the disk.
Fig. 4. Profiles of the B z Ž r . and jf Ž r .-distributions averaged over
an angle of 708. The external field Bex is successively enhanced
from ZFC state to 16 mT, 48.8 mT, 112 mT and finally to 176
mT, which is the maximum applied external field. The radial
profiles are plotted mirror-inverted for r - 0.
Fig. 3. Greyscale images of Ža. jf and Žb. jr of the disk at
Bex s 48.8 mT for partly penetrated state. In image Ža., jf ranges
from 0 Žblack. up to 2.6=10 11 Arm2 Žwhite.. The different signs
of jr in images Žb. are visualized as dark Žy. and bright Žq.
areas ranging from y1.0=10 11 up to q1.0=10 11 Arm2 .
The jr component is related to inhomogeneties like
macroscopic defects in the superconducting disk. In
Section 5, the influence of such macroscopic defects
on the current distribution in disks and squares is
described in detail.
With increasing external field, the flux penetrates
radially, starting at the disk border. In Fig. 4, the
profiles of magnetic induction Bz Ž r . and of the
angular current density jf Ž r . are plotted for different
applied external fields, increasing from the ZFC-state.
The radial profiles are averaged over an angle of 708
to suppress the noise and statistical deviations. Following the Bean model w28x, one expects a constant
critical current jc in the flux-filled part of the disk,
forming a plateau. However, as shown in Fig. 4, the
jc plateau is not spatially constant. The critical current is strongly decreased in areas of larger local flux
density, which are located mainly at the disk borders
for increasing external fields. This finding corresponds to the dependence of the critical current
density on the local magnetic flux density, which is
described in Section 4.
In addition to the jc-plateau in the penetrated part
of the disk, in a partly penetrated state we observe a
Ch. Jooss et al.r Physica C 299 (1998) 215–230
Fig. 5. Averaged profiles of B z Ž r . and jf Ž r . for decreasing
external fields demonstrating the flux reversal. The external field
Bex is successively reduced from Bex,max s176 mT to 112 mT
and 47.2 mT.
screening current js F jc in the flux-free part of the
disk. This current w25,29,30x is screening the flux-free
part of the sample inside the flux front and appears
not only for thin films but also for superconductors
with finite thickness where js flows near the surface
w31x. The screening current is zero at the disk center,
and increases monotonically until it reaches the magnitude of jc at the flux front.
Fig. 5 shows the same flux- and current density
profiles as in Fig. 4 for decreasing external fields.
When the external field is reduced from its maximum Bex,max s 176 mT, the magnetic flux density
exits through disk borders. The magnetic flux density
is dragged off, and due to the created inverse gradient in the flux-density, a zone with oppositely directed current density yjf is then penetrating. This
inverse current generates a self-field that superimposes to the self-field of the existing positive current
and the external field. The self-field of the inverse
current is visible in the negative peaks in Bz at the
disk border. The drag-off of the flux density starting
from the disk border is equivalent to the penetration
of flux-lines of opposite direction into the sample,
which annihilate with the existing flux-lines. The
annihilation takes place up to a finite penetration
223
depth, which is equivalent to the penetration depth of
the plateau of the inverse current density yjf .
The flux reversal and the penetration of an inverse
current was treated by Brandt and Indenbom w21x and
Zhu et al. w22x. They describe the current distribution
for decreasing external fields as a superposition of
two critical currents. The current distribution corresponding to the maximum external field is frozen.
Then, with decreasing external field, a new critical
state with y2 jc penetrates from the sample border,
related to an additional screening current y2 js , both
superimposing to the existing current density qjc .
This was an improvement to a previous model of
Mikheenko and Kuzovlev w25x, who neglected this
screening current. The superposition of the negative
shielding current y2 js to the existing qjc results in
a smooth crossover from yjc to qjc . This effect is
clearly visible in Figs. 5 and 2.
4. Field dependence of critical currents
Usually, the magnetic field dependence of the
critical current density of a superconductor is measured by means of a magnetometer or a transport
current as a function of the applied external field. If
the external field is much higher than the self-field
generated by the currents in the sample, one may
neglect the field screening of the sample, and the
magnetic field dependence of jc is in very good
approximation given by the dependence of the global
average current density jc on the external field. This
holds not for small magnetic fields, which will be
demonstrated in this section.
Figs. 4 and 5 show clearly that the critical current
density of the disk is not spacially constant. The
current density is decreased in regions with higher
local magnetic flux density compared to regions with
lower values of the local magnetic field. For increasing external fields, the flux density has a sharp peak
at the sample border. Therefore, the current density
is lowered towards the sample border. However, for
decreasing external fields, the situation is just the
opposite. The local magnetic field shows a negative
peak at the disk border, and the current density is
enhanced. This behaviour was qualitatively predicted
by computer simulations of the flux and current
distribution in thin films with a field-dependent criti-
Ch. Jooss et al.r Physica C 299 (1998) 215–230
224
Fig. 6. Dependence of the local critical current density on the
local flux density B z,loc for different increasing ≠ and decreasing
x external fields. The smooth solid line represents a fit of the Kim
model to the jc Ž B z,loc . data Žsee Table 1. and the dashed line
corresponds to jŽ Bex . obtained from SQUID data.
cal current density w32x, but was not measured quantitatively up to now.
An often-used model to describe field-dependencies of the critical current is the Kim model w33x. It
assumes the following field dependence of the critical current density
jc Ž B . s jc ,0
1
1 q BrB0
,
Ž 18 .
where B0 is a constant field that characterizes the
degree of field dependence and jc,0 is the current
density at zero field. In Fig. 6, the dependence of the
local critical current density on the local perpendicular component of the flux density for different increasing and decreasing external fields is plotted. All
four lines are matching approximately to a definite
and monotonous jc suppression with higher local
flux densities. Fitting the Kim model to these four
lines, one obtains B0 s 0.158 T and jc,0 s 2.98 =
10 11 Arm2 .
Let us now compare this jc Ž B . to the field dependence of the global average current density jc as
determined by SQUID measurement. Fig. 7 shows a
magnetic hysteresis loop for the disk measured at
low fields up to 200 mT Žtriangles.. The global
jc Ž Bex . can be calculated from the magnetization
hysteresis D M by w28,34x
jc Ž Bex . s
D M Ž Bex .
kR
Ž 19 .
with disk radius R and a geometry factor k s 2r3
for a disk. In Fig. 6, jc Ž Bex . is plotted as a dashed
line. It is clearly visible that the global current
density is significantly smaller than the local current
density obtained by magneto-optics. Notice, that despite this, the magnetization curve obtained by the
SQUID is quantitatively in good agreement with the
magnetization calculated from the inhomogeneous
local current density and Eq. Ž4. Ždark squares in
Fig. 7..
The reason for underestimating the jc obtained by
SQUID is the presence of macroscopic defects, which
give rise to a spatial variation of the current density.
The global current density jc derived from SQUID
measurements is an average over the local current
density. Furthermore, in the case of the SQUID
measurement one assumes that the magnetic field at
the location of the current is equal to the value of the
external field, which is strongly violated for low
external fields as demonstrated by the magnetic field
profiles visible in Fig. 4. This results in significantly
different values for B0 and jc,0 as obtained by fitting
the Kim model.
In Table 1, these values are listed for the local
field dependence, the SQUID-hysteresis shown in
Fig. 7 and additionally for another SQUID-hysteresis
at the same disk measured to a maximum external
field of 5 T. The jc,0 obtained from both SQUID
measurements is smaller compared with the jc,0 of
Fig. 7. Hysteresis of the magnetic moment m 0 M, measured by a
SQUID magnetometer for the disk. Also plotted is the magnetooptically determined m 0 M calculated from the current distribution
using Eq. Ž4.. The maximum applied fields are "200 mT for the
SQUID and 176 mT for the MOL EuSe.
Ch. Jooss et al.r Physica C 299 (1998) 215–230
Table 1
Summary of the fit-parameters jc,0 and B0 of the Kim model for
the magneto-optically determined jc Ž B z,loc . and the global jc Ž Bex .
obtained by a low-field Ž Bex,max s 0.2 T. and a high-field Ž Bex,max
s 5 T. SQUID-magnetization curve
Measurement
jc Ž Bz,loc .
D M low field
D M high field
jc,0 ŽArm2 .
11
2.98=10
1.73=10 11
1.43=10 11
B0 ŽT.
0.158
1.07
5.7
the local current density. This is due to the averaging
of jc over the macroscopic defects. The existence of
different B0 in the low and high field regimes was
also seen and discussed by other authors w35,36x, and
signals the existence of different pinning sites with
different density in the films. In conclusion, at low
external fields, i.e., if the self-field of the sample is
comparable to the external field, it is not possible to
derive the field dependence of the critical current
density from global measurements such as SQUIDmagnetization or transport current.
225
long cavity oriented parallel to the cylinder-axis,
Schuster et al. w38x pointed out that thin films offer
an enhanced sensitivity to observe such discontinuity
lines, because the magnetic flux density shows sharp
peaks at these lines. They demonstrated that the
discontinuity lines generated by a defect near a strip
or a square edge have parabolic shape. This is also
visible in simulations w8x in the framework of the
Bean model applied for a thin film with a circular
indentation. In this paper, we extend the consideration also to the discontinuity lines of a macroscopic
defect in a disk and to field dependent jc . By means
of current imaging, it is possible to image quantitatively the effect of macroscopic defects on the current, and to give a detailed description.
5.1. Square
5. Macroscopic defects in disks and squares
The effect of a macroscopic defect on the flux
density and current distribution strongly depends on
the macroscopic geometry of a superconductor.
Without any defect, the current stream lines are
defined by the condition that the distance to the
sample border has to be constant. Thus, the change
of the current flow due to a defect depends on the
Inhomogeneities or structural defects in a type-II
superconductor can have different effects on the
critical current depending on their superconducting
properties and their size. Roughly speaking, inhomogeneities on a length scale of the size of a single
vortex mainly affect the self-energy of vortices, and
therefore act as pinning sites and determine the
absolute value of the critical current density. On the
other hand, inhomogeneities on a length scale of
some vortices Žsome m m. change the current flow
also macroscopically. Campbell and Evetts w37x modeled a macroscopic defect as a cylindrical cavity
with jc s 0, which is present near the edge of a
superconductor. Since div j s 0 must be fulfilled,
macroscopic changes in the current distribution and
therefore in the flux penetration have to occur around
the cavity. They showed the appearance of discontinuity lines, where the current is sharply bending, and
which are visible as dark or bright lines in the flux
density distribution.
Whereas Campbell and Evetts regarded a long
superconducting cylinder with external field and the
Fig. 8. Flux penetration, current stream lines and current density
for a square of thickness of 300 nm and width of 2 mm in an
external field of 48 mT.
226
Ch. Jooss et al.r Physica C 299 (1998) 215–230
shape of the current stream lines in the entire superconductor.
Let us first consider defects in square-shaped thin
films. Fig. 8 shows the magnetic flux distribution of
a square as a greyscale plot. The square has a size of
2 mm = 2 mm and a thickness of 300 nm. The
applied external field is 48 mT after ZFC. In the
same figure, the calculated current stream lines are
superimposed. Several macroscopic defects at the top
and left border and the corresponding discontinuity
lines are clearly visible. For a quantitative analysis,
we confine the following discussion to the defect
placed at the left border of the sample. Fig. 9a shows
the superposition of the flux density distribution and
the current stream lines at this defect in more detail.
Both components of the current density j x Žparallel
to the edge. and j y Žperpendicular to the edge. are
plotted as a greyscale image in Fig. 9b,c, respectively. The image section represents an area of 324
= 324 m m2 . An external field of Bex s 48 mT is
applied starting from the ZFC-state.
The flux density distribution displays strong deviations in comparison with the flux penetration into a
homogeneous square. Two dark lines in the flux
density in Fig. 9a mark the positions of the bending
of the current. Whereas the penetration depth of the
flux-front is significantly enhanced in the area between the two parabolic branches of the discontinuity lines, the magnetic flux density exhibits a strong
peak at the location of the defect at the lower image
border in Fig. 9a. The current stream lines in the flux
free region in the top of this images change from a
shielding current outside to a critical current inside
the area of larger flux penetration. In the greyscale
image of j x flowing parallel to the square border
ŽFig. 9b., the defect size and the shape of the
j x-suppression is clearly visible in the current density
profile. The macroscopic defect has a size between
30 m m Žhalf depth of the j x-minimum. and 50 m m
Žonset of j x-suppression., and the j x is decreased
from 1.4 = 10 11Arm2 to 3 = 10 10Arm2 . The dark
Fig. 9. Ža. Small section of the square of Fig. 8 with a width of
324 m m. Shown are the B z distribution and the current stream
lines near a macroscopic defect. Moreover, the j x component
parallel to the film edge Žb. and the perpendicular component j y
Žc. are visible as greyscale images.
Ch. Jooss et al.r Physica C 299 (1998) 215–230
Fig. 10. Sketch of the current stream lines near a cylindrical
shaped cavity Žblack circle. in a superconducting strip or square
for jc s const. The lowest thick line represents the sample-border
and the parabola is the discontinuity line.
discontinuity lines in Bz are characterized by a local
minimum in j x and are related to the positive and
negative maxima in the j y component of the current
density, visible in Fig. 9c. These maxima mark the
positions of sharpest change in the direction of current flow. On the central line x s 0 crossing trough
the defect, j y is zero.
Fig. 10 presents a schematic drawing of the current stream lines of a cylindrical defect near an edge
of a superconducting stripe or in a square far away
from the diagonal discontinuity lines. Here, we assume a spatially constant critical current density;
hence, we regard an area of the sample where the
flux has fully penetrated and the field dependence of
jc is neglected. The parabola of the discontinuity
lines is described by r s R defrŽ1 y sin Ž f .. w38x,
where r and f are the polar coordinates and R def
denotes the radius of the cavity located at the origin.
In Cartesian coordinates one has
ys
1
2 R def
x2y
R def
2
.
Ž 20 .
Thus, it is possible to determine the defect size from
the shape of the parabola assuming a cylindrical
defect shape, jc s 0 inside the defect, and jc Ž B . s
const all over the sample.
Since one uses the parabola near the defect for
determining the defect-radius, one obtains R def s
17.7 m m; therefore, a defect size 2 R def s 35 m m.
Applying Eq. Ž20. for the parabola far away from the
defect, one gets 2 R def s 50 m m. Both values are in
good agreement with the size determined via the j x
profile. The additional widening of the parabola with
increasing distance from the defect is due to the local
field dependence of jc , which is not taken into
account in Eq. Ž20..
227
However, we would like to point out that this
geometrical derivation of the defect size is only valid
if the above-mentioned assumptions are fulfilled. On
the contrary, the determination of the defect size via
current profiles is a general method that makes no
assumptions. Additionally, it allows us to determine
quantitatively the current distribution inside the defect. The local jc and the defect size can be determined for any defect geometry, and no assumption
for jc Ž B loc . is necessary.
5.2. Disk
In order to study the effect of the macroscopic
geometry of the superconductor on the current modification by a macroscopic defect, we consider now
the disk geometry. Fig. 11a shows the flux density
distribution of a section of the disk visible in Fig. 2.
The section containing a macroscopic defect has a
size of 453 = 453 m m2 . The external field is Bex s
176 mT applied to the ZFC-state. The current stream
lines are superimposed to the greyscale plot of Bz .
The shape of the dark discontinuity lines in Bz
extending from the defect towards the disk center is
strongly different from the shape in a square. Even
the lines start to run at the defect location as an
approximate parabola, towards the disk center, they
change their shape into almost straight lines; afterwards, the two branches approach and meet each
other near the disk center. In addition there are
further discontinuity lines, which are related to other
macroscopic defects and force a strongly inhomogeneous flux front near the disk center.
The macroscopic defect suppresses jf from about
1.9 = 10 11 to 0.3 =10 11 Arm2 . The size can be
determined from the jf-profile in Fig. 12b and is
between 49 m m Žhalf-depth of the jf-minimum. and
88 m m Žonset of the jf-suppression.. The jf -component exhibits a local minimum along the discontinuity lines, while the jr component of the critical
current, which is visible as a greyscale image in Fig.
11c, exhibits local positive maxima and negative
minima.
In Fig. 12, the current stream lines are sketched
for a disk containing a cylindrical cavity with jc Ž B .
s const ŽBean model. all over the sample. From the
228
Ch. Jooss et al.r Physica C 299 (1998) 215–230
conditions of equidistance of the stream lines Ž jc s
const. and div j s 0, one can construct geometrically
the shape of the discontinuity line. In polar coordinates, the discontinuity line is given by
2
2 RR 0 cos Ž f y f 0 . y R 2 y R 02 q R def
r Ž f . sRy
.
2 R 0 cos Ž f y f 0 . y 2 R y 2 R def
Ž 21 .
The position of the center of the cylindrical cavity
with radius R def is characterized by the radius R 0 to
the disk center and the angle f 0 . The radius of the
disk is R. Note that the shape of the closed discontinuity line, given by Eq. Ž21., depends also strongly
on the radial position of the macroscopic defect R 0
and not only on the defect radius R def , and the
geometrical relation between the defect size and the
shape of the discontinuity line is much more complex than in a square.
Let us now compare the result of this Bean-like
model with the experimental observation. In contrast
to the model depicted in Fig. 12, the current stream
lines near the defect in Fig. 11a are bending as well
towards the disk center as towards the disk border.
This is visible in the positive curvature of the current
stream line below the macroscopic defect. This is
also visible in the additional .jr-structure in the area
Aqs R 0 q R def - r - R, f f f 0 4 between the defect and the disk border. Inside Aq the radial component of the current density has an inverse sign in
comparison to the "jr structure in the area Ays 0
- r - R 0 y R def , f f f 0 4 between the macroscopic
defect and the disk center, which is related to the
discontinuity lines.
Within the Bean model Ž jc Ž B . s const., the current stream lines cannot bend towards the disk border inside Aq and one has only a "jr-structure
inside Ay. According to Ampere’s
law Bz s Hrjr d f
`
this " jr-structure is related to the positive peak in
the flux density distribution at the macroscopic defect. However, there is an additional dark spot in the
flux density in Fig. 11a at the defect location corresponding to a local minimum in Bz .
Fig. 11. Ža. Greyscale image of the flux density distribution
together with the current stream lines near a macroscopic defect in
the disk. Shown is a section of the disk of 453=453 m m2 at
Bex s176 mT. Also plotted are the two current density components Žb. jf and Žc. jr for the same section of the sample.
Ch. Jooss et al.r Physica C 299 (1998) 215–230
229
A basic requirement for this effect is that the
macroscopic defect is located sufficiently far away
from the sample border. This ensures a large field
gradient between the defect and the border, and a
significant jc Ž Bz . can turn up in this area.
6. Conclusions
Fig. 12. Sketch of the current stream lines in a superconducting
disk with jc s const, containing a cylindrical shaped cavity Žblack
circle.. The thicker line of approximately elliptical shape represents the discontinuity line of the currents.
We suggest that this characteristic difference to
the Bean-like model is a result of the strong localfield dependence of jc . Note that the local minimum
in Bz is related to a maximum in the angular current
density jf , visible as a bright spot near the defect in
Fig. 11b. This maximum of jf has the same magnitude as the current density at the flux front Ž Bz s 0.
near the disk center.
Further, we consider now the modifications to the
Bean-like model due to a local-field dependent jc :
Within the Bean-model Ž jc s const., the flux density
at a position r 1 s Ž R 0 q R def q e , f 0 . directly near
the macroscopic defect is lower than the flux density
at a position r 2 s Ž R 0 q R def q e , f 4 f 0 . at the
same radius but far away from the defect. The
parameter e is small comparing to the defect radius
R def . If we start our discussion with this initial stage,
which is approximately realized for small external
fields and now allow a local field-dependence of jc ,
the current density is inevitably larger near the defect
at r 1 than far away at r 2 . Consequently, the current
is slightly bending towards the disk border. The
related inverse .jr-structure is then generating a
local minimum in Bz . This effect is self-amplifying
during the flux penetration with increasing Hex ,
since the local-field dependence of jc is increasing
with larger flux density gradients and jf Ž Bz . in the
area Aq should be maximized.
The presented exact and very fast inversion
scheme gives new insight into the details of flux
penetration and related local critical current density
in type-II superconductors with strong pinning. The
local current density in the film plane are directly
imaged with high resolution of 6 m m and an accuracy of 20% up to 5% depending on the case. The
local critical current density varies inside the specimen due to geometry-, defect- and field-dependence
of jc . Therefore, the average jc determined from
magnetization measurements usually underestimates
the true local jc flowing in the sample. We proved
that at low external fields F 200 mT, the critical
current density depends on the local field. This local
field dependence for small fields is much stronger
than the usually determined external field dependence of the average jc determined by magnetometer.
This deviation is due to the large field gradients in
the superconductor with partly penetrated flux.
We quantitatively imaged the local currents near
macroscopic defects. Due to the nonlocal field–current relationship, this requires a method that images
the current of the sample as a whole, and enables a
high resolution at the same time. The influence of
macroscopic defects on the current density in the
YBaCuO thin films is shown in detail, depending on
sample geometry and jc Ž Bloc .. For the disk, we
found significant deviations from the predictions of a
Bean-like critical state model.
The fast inversion of measured magnetic field-data
allows a detailed quality analysis of the homogeneity
of a superconducting film and a characterization of
the size and the jc-suppressions of the macroscopic
defects. The results show that critical current densities determined by global measurement methods like
magnetization or transport current measurements
must be interpreted very carefully if little is known
about the defects in the superconducting material.
230
Ch. Jooss et al.r Physica C 299 (1998) 215–230
Acknowledgements
The authors wish to thank T. Dragon for the
evaporation of the magneto-optical layers, H.-U.
Habermeier and B. Leibold from Technology-Group
of the Max-Planck-Institut fur
¨ Festkorperforschung
¨
for the fabrication and chemical etching of several
YBaCuO thin films, and E.H. Brandt for intensive
discussion and helpful advice on the problems of
inversion of Biot–Savart’s law.
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