Physica C 312 Ž1999. 247–254
Magneto-optical observations of magnetic flux distribution
in a high-temperature superconductor x-array
Z.W. Lin
a,1
, J.W. Cochrane a , N.E. Lumpkin b, G.J. Russell
a,)
a
b
AdÕanced Electronic Materials Group, School of Physics, UniÕersity of New South Wales, Sydney NSW 2052, Australia
Semiconductor Nanofabrication Facility, School of Physics, UniÕersity of New South Wales, Sydney NSW 2052, Australia
Received 7 September 1998; revised 3 December 1998; accepted 8 December 1998
Abstract
The magnetic flux distributions in samples consisting of five parallel strips of NdBa 2 Cu 3 O 7y d thin film arranged in an
x-array, were visualized using a magneto-optical technique. The external magnetic field was applied perpendicular to the
surface of each sample held at 40 K. The spatial magnetic flux profiles were also theoretically calculated taking into account
the thickness of the sample and the distance between the surface of the sample and the indicator film. The experimental
results showed excellent agreement with the calculated flux distribution and showed that for the central strip, flux penetrates
deeper into the strip than for the case of a single isolated strip. This is due to field compression resulting from the shielding
of the superconducting strips on each side. q 1999 Published by Elsevier Science B.V. All rights reserved.
Keywords: Magneto-optical; Magnetic flux; HTSC
1. Introduction
There have been numerous experimental and theoretical studies of magnetic flux and current distributions in high-temperature superconductors ŽHTSCs.
w1–10x. The experimental investigations have involved a number of techniques, such as a scanning
Hall probe w11x, Hall array w12x, decoration technique
w13x, scanning superconducting quantum interference
device microscopy w14x, magnetic-force microscopy
w15x, and the magneto-optical ŽM-O. imaging method
w16x. On the theoretical side, a variety of methods
)
Corresponding author. Tel.: q61-2-9385-4561; Fax: q61-29385-6060; E-mail: g.russell@unsw.edu.au
1
E-mail: lzw@newt.phys.unsw.edu.au.
have been used to calculate the flux distribution.
These include conformal mapping w1x, numerical
methods w2,3x and analytical calculations w4x. Different sample geometries have also been studied w5–10x,
such as a strip, rectangle, triangle, disk, cylinder,
hollow cylinder and a slab. Among these geometries,
thin-film strips are the most significant, not only for
their possible applications but also for their simplicity in theoretical and experimental investigations.
Recently, the thin-film strip has been investigated for
passive microwave applications and communications
w17,18x, while there have been significant developments in the theory of the statics and dynamics of
vortices in such thin films w12,16x. These studies
were based on isolated strips. A more challenging
and practical geometrical pattern involves thin film
strips arranged in an x-array or z-stack.
0921-4534r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 6 9 5 - 9
248
Z.W. Lin et al.r Physica C 312 (1999) 247–254
The electromagnetic properties of high-temperature superconducting thin-film strips arranged in an
x-array or z-stack have recently been theoretically
investigated by Mawatari w19x. The magnetic field
and current distributions at the surface of samples
containing both geometries, in a perpendicular external field, were calculated using a transformation
w20x followed this work with an
method. Muller
¨
analytical study involving a transport current passing
through similar geometrical strip arrangements. These
calculations assumed that the thickness of each thin
film strip was negligible.
Among the experimental techniques, the M-O
method is the most powerful in providing detailed
information on local flux dynamics at any desired
temperature. Moreover, this technique can also provide information on the pinning force w21,22x and
critical current distribution w23x. In this technique,
the measured flux distribution is observed in an iron
garnet indicator film, which is at some distance
above the sample surface.
In this report, we first derive the integral equation
for the spatial flux distribution at some distance
above the surface of a finite thickness thin film
sample formed into an x-array arrangement. Then
the measured magnetic flux distributions are determined by the M-O technique for an x-array system
fabricated from a high-Tc film in a perpendicular
external field at a temperature of 40 K. The x-array
consisted of five parallel thin-film strips involving
two geometrical factors, LrW s 3 and 4, where L is
the distance between the centres of adjacent strips
and W is the half width of a strip. Hence, we can
investigate the strength of the interactions between
adjacent superconducting strips.
2. Theory
In this section, we present the spatial flux distribution profiles at some distance D above the surface
of a finite thickness sample for an x-array arrangement, which is defined as an infinite number of
parallel strips aligned in the x–y plane and at regular
intervals L between adjacent strips. The width of
each superconducting strip is 2W while their thickness d < W. The nth superconducting strip occupies
an area where < x y nL < - W Ž n s 0, "1, "2, . . .
"`.. < y < - `, < z < F dr2. It is assumed that the
critical current density Jc is constant for each strip,
that is, independent of the local flux density. When
the external magnetic field Ha is applied parallel to
the z-axis, the current flowing in each strip along the
y-direction, J Ž x ., and the perpendicular component
of field at the sample surface, H Ž x ., are given by
Mawatari w19x as:
2 Jc
JŽ x. s
p
J Ž x . s Jc
< x q nL < - a
arctan w Ž x . ,
x
< x<
,
a - < x q nL < - W
Ž 1.
< x<-a
H Ž x . s 0,
H Ž x . s H0 arctanh Ž 1r< w Ž x . < . ,
H Ž x . s H0 arctanh < w Ž x . < ,
a - < x q nL < - W
< x q nL < ) W
Ž 2.
where
H0 s Jc drp ,
wŽ x. s
tan Ž p xrL .
tan Ž p WrL .
(
=
as
L
p
arcsin
ž
tan2 Ž p WrL . y tan2 Ž p arL .
<tan2 Ž p arL . y tan2 Ž p xrL . <
sin Ž p WrL .
sin Ž HarH0 .
/
,
,
Ž 3.
Ž 4.
where a donates flux front measured from the centre
of the strip. We note that these equations do not
include the variables d and D for the sample thickness and distance between the sample surface and
indicator film, respectively.
Since J Ž x . and H Ž x . are periodic with period L
as J Ž x q nL. s J Ž x . and H Ž x q nL. s H Ž x ., it is
sufficient to consider only the field for < x < - Lr2.
Fig. 1 shows the profiles of magnetic field H Ž x . and
current density J Ž x . at the sample surface of an
x-array arrangement in which LrW s 3 and the
dashed line is for an isolated strip.
As the thickness d of the superconducting thin
film is less than the penetration depth of the magnetic field, it is reasonable to consider that the
current distribution is independent of the variable z,
Z.W. Lin et al.r Physica C 312 (1999) 247–254
249
perpendicular to the film surface. Thus, the field
distribution at some distance, D, above the sample
surface for an x-array with a film thickness, d, can
be calculated by inserting the current distribution,
Eq. Ž1., into the Biot–Savarts law. The normal component of the field Hz Ž x . is therefore given by:
Hz Ž x . s
1
2p
=
ns`
Ý
nsy`
dr2
wqnL
HywqnLd x
X
J Ž x . Ž x y xX .
Hydr2 Ž x y x .
X 2
X 2
q Ž Dyz .
d zX q Ha .
Ž 5.
Fig. 2 shows the profiles of the normal component of
the magnetic field at different distances, D, above
the sample surface for different applied fields, Ha .
One can see that the magnetic field profiles become
smoother with increasing distance. The gradient of
the field at the edge of the strip and the extent of
Fig. 2. Field profiles from a numerical calculation using Eq. Ž5.
for an x-array with Lr W s 3, ds 200 nm at Ds 3 mm, 25 mm,
50 mm. Ha r H0 is Ža. 0.5, Žb. 3 and Žc. 10.
shielding increases with increasing of the product
Jc d.
3. Experimental
Fig. 1. Profiles of Ža. magnetic field calculated from Eq. Ž2. and
Žb. current density from Eq. Ž1. for an x-array with Lr W s 3
Žsolid lines. at Ha r H0 s 0.5, 3, 10. The dashed lines are for an
isolated strip. The strip is located at y1F x r W F1.
In this study, c-axis-oriented NdBa 2 Cu 3 O 7y d
ŽNBCO. thin films were prepared by the pulsed laser
deposition technique onto 10 = 10 mm2 SrTiO 3 substrates. The films had a thickness d s 200 nm and a
critical current density Jc s 3 = 10 6 Arcm2 at 40 K.
The thin films were patterned using a standard lithographic technique. Sample A had a geometrical factor of LrW s 3 with L s 0.6 mm and W s 0.2 mm
while sample B had LrW s 4 with L s 0.8 mm and
W s 0.2 mm. The length of both samples, H, was 10
mm with ratios Hr2W s 25 and 2Wrd 4 1.
The principle of the M-O technique used in these
measurements is based on the Faraday effect. When
250
Z.W. Lin et al.r Physica C 312 (1999) 247–254
Z.W. Lin et al.r Physica C 312 (1999) 247–254
plane polarized light is transmitted through a magneto-optical material in a direction parallel to a
magnetic field, the plane of polarization is rotated. In
this investigation, a Bi-doped iron garnet film with
perpendicular magnetization was used as an active
indicator film. The film was prepared on a transparent ŽGdCa. 3 ŽGaMgZr.5 O 12 substrate by the liquidphase epitaxy method w24x. In the iron garnet film, a
labyrinth domain structure with a black and white
stripe is observed in polarized light w25x. In zero
field, the domain width for one direction of magnetization is equal to that for the opposite direction.
When a magnetic field is applied perpendicular to
the film surface, the domain width with the magnetization parallel to the field expands and the one with
the opposite magnetization becomes narrower, so the
flux distribution can be obtained by analyzing the
domain structure. Two types of iron garnet film were
used in this experiment, a brown one which has a
high saturation field and a yellow one which is very
sensitive to magnetic fields. For each film the relationship between the field and domain width is nonlinear and dependent on temperature as shown by
Lin et al. w26x.
The M-O system consisted of a solenoid cooled
by liquid nitrogen, an optical cryostat, a polarization
microscope and a CCD camera. The specimen was
placed on a thin insulating but highly thermal conducting sapphire substrate which was directly positioned on the cold finger inside the optical cryostat.
In this configuration, the specimen can be cooled
down to 40 K. In order to observe a larger sample
than the field of view of the microscope, the polarization microscope was installed on an x–y stage.
The images of magnetic flux patterns were observed
using the microscope in the reflection mode making
use of the double Faraday effect and were transferred
to a PC via the CCD camera. Further details of the
experimental system have been given by Lin et al.
w26x.
The superconducting thin film arrays were zero
field cooled ŽZFC. to 40 K, then an external mag-
251
Fig. 4. Comparison of experimental data Ždots. obtained from Fig.
3a,b and c along the line AB in Fig. 3 with calculated field
profiles Žsolid lines. at m 0 Ha s 343 G, 365 G and 374 G Žfrom
bottom to top. taking into account film thickness ds 200 nm, and
the distance Ds 40 mm above the sample surface using Eq. Ž5..
Here, Ls 0.6 mm, Jc s 3=10 6 Arcm2 . The middle strip is
located at y0.2 F x F 0.2.
netic field was applied and increased to the maximum of 800 G perpendicular to the surface of the
thin film Žhence parallel to the c-axis., then, decreased to zero. Photomicrograph images were stored
as significant changes occurred.
4. Result and discussion
Fig. 3 shows the changes of the stripe domain
patterns in the brown iron garnet film for sample A
Ž LrW s 3., zero field cooled to 40 K, with an
increasing external magnetic field. In Fig. 3, the
black vertical lines indicate the edges of the HTSC
thin film strips and the number of each strip is
marked. The black spots are caused by defects in the
indicator film and CCD camera. The fields at the left
and right sides of the sample monotonically decrease
with distance from the two side strips. The black
horizontal line AB indicates the line for which the
Fig. 3. Variation of stripe domain patterns in the brown indicator film for sample A after zero field cooling to 40 K. Application of an
external magnetic field increasing for Ža. m 0 Ha s 343 G, Žb. 365 G and Žc. 374 G up to 800 G, then down to Žd. 346 G and Že. 327 G. The
black vertical lines indicate the edges of the thin film strips and the black horizontal line indicates the line along which the field pattern was
measured for Ža., Žb. and Žc. with the results being shown in Fig. 4. The number of each strip is marked.
252
Z.W. Lin et al.r Physica C 312 (1999) 247–254
field profiles were measured and the results are
shown in Fig. 4. The fields in the centre of each strip
are much smaller than the applied fields and the
magnetic field gradients are clearly seen over the
entire sample A for different fields. When the applied field is increased, the stripe patterns show
evidence that the magnetic flux gradually penetrates
from the edges of each strip into the centre. However, the fields in the gap Žbetween the strips. were
much larger than the applied field. This can be
explained by field compression in the gap due to the
shielding of the superconducting strips on each side.
Thus, if the strips were closer, the fields in the gaps
would be considerably stronger and the flux would
penetrate deeper into the strips arranged in an x-array
than for the case of an isolated strip at the same
applied magnetic field.
Comparing the measured field profiles for sample
A along the line AB with the analytically calculated
results at the surface of the thin film for the x-array
arrangement, the general agreement is good, but the
measured flux profiles are smoother than the theoretical ones. This distinction has also been found in the
case of an isolated strip w3x. However, the experimental profiles show excellent agreement with the results
of a numerical calculation using Eq. Ž5. in which the
thickness of the thin film and distance between the
sample surface and the indicator film are taken into
account. The parameters for the sample are Jc s 3 =
10 6 Arcm2 , d s 200 nm and W s 0.2 mm, however,
the distance D between the indicator film and superconducting film surface cannot be experimentally
determined. After trying different values of D, we
found that for D s 40 mm, the experimental profiles
Fig. 5. Variation of stripe domain pattern in the yellow indicator film for sample B Žafter zero field cooled down to 40 K.. Application of an
external field increasing for Ža. m 0 Ha s 21 G, Žb. 28 G and Žc. 38 G, up to 800 G, then down to Žd. zero. The white horizontal line AB
indicates the line along which the field pattern was measured and the results are shown in Fig. 6. The number of each strip is marked.
Z.W. Lin et al.r Physica C 312 (1999) 247–254
agreed extremely well with the numerical calculation. Fig. 4 shows this good agreement for the
middle strip, in which the solid curves are obtained
by substituting the above parameters into Eq. Ž5.; the
dots are the experimental data.
The variation of the domain patterns for sample A
Žzero-field cooled to 40 K. were also taken in decreasing external field after a maximum field of 800
G had been applied. As sample A is larger than the
field of view of the microscope and the field distribution is periodic, the middle section of the domain
pattern, that is, for strips 2, 3, and 4, are shown in
Fig. 3d and e. At a field of 346 G, the stripe domain
pattern appeared in the gap, as well as at the edges of
each strip, and just started moving into the centre of
the strip. The flux trapped in each strip was expelled
first at the strip edge. Decreasing the field further,
the stripe domain moved towards the centre of the
strip, as flux was further expelled.
A similar behaviour was also observed for an
isolated thin film strip when the external field was
applied perpendicular to the surface of the sample
w27x. Fortunately, the theoretically calculated flux
profiles for an isolated strip and an x-array arrangement w6,19x, shown in Fig. 1, show the similarity of
the flux distributions inside the strips though there is
a significant difference outside the strip as discussed
above.
Fig. 5 shows the variation of stripe domain patterns in the high sensitivity yellow indicator film for
sample B Ž LrW s 4, zero-field cooled to 40 K. after
the application of an increasing and decreasing perpendicular magnetic field. For the reasons stated
above, only the domain patterns for the middle part
of the sample are shown. In Fig. 5, the black and
white spots are defects in the yellow film and CCD
camera while the white vertical lines indicate the
edges of the thin film strips. In this part of the
experiment, the high sensitivity indicator film was
employed to visualize the flux distribution for weak
magnetic fields. At low external fields, the flux
penetrates into the edge of each strip with a significant gradient towards the centre. Hence, the inductive currents flowing in these regions shield the
centre of the strip and maintain the centre free of
magnetic flux. Increasing the field, the flux fronts
reach the centre of each strip and the strip is eventually fully penetrated by the magnetic field. Decreas-
253
Fig. 6. Comparison of experimental data Ždots. obtained from Fig.
5a,b and c along the line AB in Fig. 5 with calculated field
profiles Žsolid lines. at m 0 Ha s 21 G, 28 G and 38 G Žfrom
bottom to top. taking into account film thickness ds 200 nm, and
the distance Ds 40 mm above the sample surface. The parameter
values for sample B are Jc s 3=10 6 Arcm2 and Ls 0.8 mm.
The middle strip is located at y0.2 F x F 0.2.
ing the field after a maximum of 800 G, the trapped
flux in the strip is expelled. For the remanent state
some flux was pinned in the centre. Comparing the
results for samples A and B, one can see that the
behaviour of the flux in the strips is similar while the
field characteristic in the gap between the strips are
also analogous, although there is an observed increase in the magnetic field for LrW s 3 compared
to that for 4.
Comparing the field distributions corresponding
to Fig. 5a, b and c along the white horizontal line
AB for the middle, number 3 strip, with the theoretical Eq. Ž5., as shown in Fig. 6, we find that the
experimental profiles show excellent agreement with
the theoretical field profile distribution at 40 mm
above the sample surface. However, the field to the
right of the middle is slightly stronger than to the
left, which is also observed in Fig. 4. We assume
that it is caused by the applied fields being slightly
inhomogeneous over the width of the sample.
5. Conclusion
The characteristic behavior of magnetic flux in
thin film strips arranged in an x-array has been
directly observed using the magneto-optical tech-
254
Z.W. Lin et al.r Physica C 312 (1999) 247–254
nique and an iron garnet film with perpendicular
magnetization and a stripe domain pattern at 40 K,
upon the application of a perpendicular magnetic
field. The observed experimental results show excellent agreement with numerical calculations if the
distance between the surface of the specimen and the
iron garnet indicator film, and the thickness of the
sample itself are taken into account. Furthermore, the
flux penetration for the centre strip is deeper than
that for an isolated strip at the same applied field
because of field enhancement at the edges of the
strip due to the x-array arrangement.
Acknowledgements
Z.W. Lin gratefully acknowledges the Australia
Research Council for an Australian Postgraduate
Award ŽAPA., and thanks Professor H. Kronmuller
¨
for valuable discussions. Z.W. Lin appreciates encouragement and assistance from Dr. Jian of the
School of Chemistry, University of New South
Wales.
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