PHYSICAL REVIEW B
VOLUME 56, NUMBER 9
1 SEPTEMBER 1997-I
Meissner holes in superconductors
V. K. Vlasko-Vlasov
Institute of Solid State Physics RAS, 142432 Chernogolovka, Russia
and Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439
U. Welp, G. W. Crabtree, and D. Gunter
Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439
V. Kabanov and V. I. Nikitenko
Institute of Solid State Physics RAS, 142432 Chernogolovka, Russia
~Received 13 February 1997!
Features of the magnetic induction patterns at the boundary between oppositely magnetized parts in YBCO
crystals are studied using an advanced magneto-optical technique. Vortices of opposite polarities are revealed
to concentrate at the remagnetization front, equivalent to a local increase of the current along it. A model is
suggested considering the presence at the front of a flux-free cylinder formed by closed vortex loops. Inside the
cylinder the loops collapse because their line tension exceeds the pinning force. Currents along the cylinder
surface and in its nearest environment can be large and determine observed flux patterns. A role of the
Meissner holes in the appearance of macroturbulent flux instabilities is discussed. @S0163-1829~97!02633-7#
I. INTRODUCTION
Remagnetization of type-II superconductors is usually
discussed in terms of the one-dimensional critical-state
model1 supposing samples infinite in the field direction
~plates or cylinders! with straight vortices parallel to the
field. The remagnetization front is then a surface separating
vortices of opposite polarities and parallel to them. In the
present work it is shown experimentally that remagnetization
of superconducting plates in normal fields takes place by
shrinking of closed induction loops centered on the front line
rather than by the motion of a flat boundary. Observations of
the magneto-optical images on the faces of YBCO single
crystals show that magnetic flux is concentrated at the remagnetization front which implies a local increase of the
current along it and the formation of concentric vortex loops
around the front line. Thus the front has a specific threedimensional ~3D! structure in the plates. A model is discussed suggesting that even in a superconductor fully in the
critical state there is a flux-free cylinder conferred on the
front whose radius is determined by the equilibrium between
the vortex line tension and the pinning force. We call these
vortex-free regions Meissner holes. They determine the scenario of remagnetization by asymmetric collapse of inner
vortex loops. Estimates and numerical fits of measured induction profiles show that in anisotropic superconductors an
important contribution to the local current along the front is
associated with anisotropic critical currents in the close environment of the Meissner hole. The total excess current in
the Meissner hole results in dynamical instabilities showing
themselves in meandering the front and formation of vortex
bundles ~macroturbulent current structures! as observed in
Ref. 2.
II. RESULTS AND DISCUSSION
The remagnetization patterns were studied on different
faces of high-quality YBCO single-crystal plates (T c ;93 K,
0163-1829/97/56~9!/5622~9!/$10.00
56
DT;0.2 K! in an applied field perpendicular to the ab
plane. Crystals with width-to-thickness ratios from 625/110
to 1020/25 mm were imaged. Dimensions of the samples
illustrating the main observed peculiarities are shown in the
figure captions. Both densely twinned and nearly untwinned
crystals were investigated. Flux distributions were imaged
using an advanced magneto-optical technique.3 This technique yields pictures of the field penetration and provides
measurements of the normal component of induction on the
sample surface.
A. ab-face observations and induction profiles
Figure 1 shows typical magneto-optical images at temperatures of 20 K @Figs. 1~a! and 1~b!# and 55 K @Figs. 1~c!
and 1~d!# in a 40-mm-thick YBCO crystal during remagnetization. A positive magnetic field was increased to 1.5 kOe,
then decreased to zero @Figs. 1~a! and 1~c!# and a negative
field @ H521000 Oe in Fig. 1~b! and 2230 Oe in Fig. 1~d!#
was applied. In these pictures taken with crossed polaroids, a
brighter image intensity corresponds to a higher normal component of induction (B n ) and the darker one to lower B n .
The remagnetization front is revealed in Fig. 1 as a dark line.
At H50 the negative flux due to demagnetizing fields of
trapped vortices starts entering near the sample edges @Figs.
1~a! and 1~c!#. Numerical calculations for the perpendicularly magnetized superconducting plates4,5 suggest such a behavior. With increasing negative field the boundary between
up- and down-polarized vortices moves inside the crystal
@Figs. 1~b! and 1~d!#. At lower temperatures @Figs. 1~a! and
1~b!# it is practically parallel to the sample edges and bends
slightly towards corners. At higher temperatures @Figs. 1~c!
and 1~d!# the remagnetization front meanders and vortices
gather on it. This is revealed in Figs. 1~c! and 1~d! by increased intensity adjacent to the front.
Profiles of the normal component of induction ~Fig. 2!
along lines crossing the sample as shown in Fig. 1 reveal
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© 1997 The American Physical Society
56
MEISSNER HOLES IN SUPERCONDUCTORS
5623
FIG. 1. Magnetic flux patterns in a 40-mm-thick normally magnetized YBCO single crystal. ~a! Trapped flux after cooling in 1.5 kOe and
switching off the field at 20 K. The boundary between up ~s! and down ~ % ! magnetized areas is close to the crystal edges. Outside the
sample there is a ‘‘negative’’ stray field of the trapped flux. ~b! At 20 K, a negative field of 21 kOe is applied after ~a!, and the
remagnetization front moved further inside the sample. ~c! Trapped flux after application and switching off 1.5 kOe at 55 K; the boundary
between two flux directions is narrow and strongly bent. ~d! 55 K, H52230 Oe after ~c!.
essential changes of the flux distribution near the remagnetization front at higher temperatures. At lower T the profiles
are smooth and peculiarities in the front region are hardly
observed ~Fig. 2, left half!. With increasing temperature a
sharp kink and at T>40 K a zigzag curving of B n (x) ~Fig. 2,
right half! appears around the zero B n level. Such a peculiarity on the profile is similar to the concentration of opposite
fields around a current-carrying wire and implies a local increase of the current along the front.
B. Basic model
FIG. 2. Profiles of the normal component of induction measured
along the line shown in Fig. 1~c!. Halves of scans at 20 and 40 K
are shown.
This local increase of the current cannot be explained by a
field dependence of the critical current ~numerical calculations of this effect predict gradual changes of J c and B n at
the front6! and should be associated with a specific vortex
and current structure formed at the remagnetization front.
The discussion below is based on the model developed independently by Koppe7 and Evetts and Campbell8,9 for current
flow in a superconducting wire containing closed vortex
loops. This model was also used in Ref. 10 for explanation of
some variations of B n profiles measured on the ab surface in
BSCCO crystals.
The main features of the Meissner hole and the surrounding vortex loops are shown in Fig. 3~a!. There will be a
minimum radius of the loop, R M , determined by the balance
of the line tension force F t 5 s /R M ~in the isotropic case the
line tension s equals the vortex line energy «5B c1 F 0 / m 0 ,
where F 0 is the flux quantum! and the pinning force F p . At
small fields F p 5J c F 0 and R M 5B c1 /( m 0 J c ). Loops of
smaller radius will collapse as for them F t .F p . As a result
a cylindrical flux-free region of diameter 2R M should be
5624
V. K. VLASKO-VLASOV et al.
56
plate in the perpendicular field. Figure 4~a! shows a distribution of field lines calculated for an infinitely long bar with
the width-to-thickness ratio close to that of a real sample
shown in the next pictures. The picture simulates the critical
state in the decreasing field normal to the wide plane of the
bar. A similar flux distribution was obtained by Brandt using
strict electromagnetic calculations of the field penetration in
the transverse geometry.18 The density and direction of the
curved lines in Fig. 4~a! correspond to the strength and direction of the field. Currents of a constant density and opposite directions are supposed to flow in the left and right
halves of the bar parallel to its length. One can see that stray
fields of the trapped flux form closed induction loops near
the sample edges at decreasing field. This structure should
move to the interior at switching off and application of the
negative field and the closed vortex loops will form inside
the sample. So Meissner holes should appear first near the
end face of the plate parallel to the field direction.
C. Contemporary ab and end-face observations
FIG. 3. Schemes of closing vortex loops ~a! and the formation
of the Meissner hole at the remagnetization front ~b!. Critical currents J c in the bulk and the Meissner current I M along the boundary
of the flux-free cylinder are flowing in the same direction.
formed7–9 along the front @Fig. 3~b!# and the Meissner current of density J M ;(1/m 0 )B c1 /l should flow in the l layer
along its surface. Such a current along the boundary between
the Meissner and mixed phases is usually discussed in terms
of the dB/dH effect.9,11–15 This suggests a specific shape of
the B(H) curve for type-II superconductors near the first
critical field where the magnetization current riches a maximum. The integral Meissner current i M 52 p R M lJ M
52 p (B c1 / m 0 ) 2 /J c flowing in the same direction as J c in the
environment is twice larger than the loss of the critical current p (R M ) 2 J c 5 p (B c1 / m 0 ) 2 /J c inside the flux-free cylinder. As a result, the excess local current 0.5i M produces an
appropriate distortion of the flux profile near the B n 50 level.
The value and distance between the extrema of the field
variation due to the excess current are inversely proportional
to the distance of the current line from the surface. The
changing flux concentration at the remagnetization front and
the front image width in Figs. 1~c! and 1~d! can be associated
with wiggling of the Meissner hole along the thickness of the
sample.
Using the above model the stronger effect of the Meissner
holes at higher temperatures ~Figs. 1 and 2! can be explained
by an increased excess current value. It increases with dimensions of the hole given by B c1 /J c ratio. At heating the
sample J c drops much faster @in YBCO as exp(2T/T0) ~Ref.
16!# than B c1 „which changes as ;l 22 ; @ 12(T/T c ) 4 # ~Ref.
17!… so that i M ;(B c1 /J c )B c1 and the distortion of B n (x)
can increase. This distortion should be more pronounced on
the lower general slope of the induction profile determined
by decreased J c . Also, at higher T bending of the Meissner
holes increases so that they can come closer to the surface
and give a stronger flux perturbation there. These same factors operate in the anisotropic case discussed below.
The formation of the Meissner holes is a natural consequence of changing the field distribution at the edge of a
To check the above suggestion, flux distributions on the
end faces of the samples were studied in a field normal to the
ab plane. Figure 4~b! shows a flux image on the end face in
a 110-mm-thick crystal observed after application of 11.8
kOe and decreasing the field to zero. A concentration of
positive and negative B n components at the long sample
edges ~bright and dark contrast! is due to demagnetizing
fields of the trapped flux. A peculiarity similar to that observed at the remagnetization front on the ab plane and demonstrated in Fig. 1 is distinctly seen along the horizontal
median of the ac face. This meandering structure with a
local concentration of opposite normal field components reveals the formation of a Meissner hole near the end face.
Similar to the above-discussed ab patterns, the Meissner
hole image on the end face @Fig. 4~b!# has more bright and
less bright segments. They are associated with parts of the
excess current line closer to and further from the surface and
reveal a bending of the Meissner hole in the ab direction. In
turn, a wiggling of the Meissner hole image on the end face
confirms the above suggestion that the Meissner hole bends
also along the sample thickness. After application of the
negative field, the structure disappears @Fig. 4~c!#. As follows
from ab-face observations in the same conditions @Figs. 4~d!
and 4~e!#, this corresponds to a shift of the Meissner hole
from the edge to the interior of the sample. Similar observations in thinner samples are illustrated by Fig. 5.
Profiles of the normal component of induction measured
across the end face @Fig. 4~f!# show anomalous perturbations
of B n (x) with the same features and dimensions close to
those measured at the remagnetization front on the ab surface of thin samples.
It should be noted that features due to the Meissner holes
on the end face develop in some delay time ~decreasing with
temperatures! after switching off the field. This indicates that
closure and collapse of vortices resulting in the formation of
Meissner holes is a dynamical process occurring through flux
creep. After formation, the image of the Meissner hole
smears with time, indicating that it moves deeper into the
sample, as expected for a remagnetization front in a sample
with trapped flux at zero field.4–6
56
MEISSNER HOLES IN SUPERCONDUCTORS
5625
FIG. 4. Appearance of the Meissner hole on
the end face of a normally remagnetized plate: ~a!
field lines tilting at decreasing external field calculated for a long plate with opposite constant
density currents in two halves and ~b! end-face
magneto-optical image of the trapped flux in a
110-mm-thick YBCO crystal at 53 K (H50 after
application of 1.8kOe i c). ~c! At H52170 Oe
the Meissner hole moved inside the sample and
its image washed away. ~d! and ~e! ab-face images in the same conditions as ~b! and ~c!, respectively. ~f! Flux profile across the sample in ~b!.
Fitting curve in ~f! is calculated at 2 mm ~in accordance with the indicator film thickness! above
infinitely long 110-mm-thick and 580-mm-wide
plate with J c 55.253104 A/cm2 in the bulk and
J M 56.83106 A/cm2 in a 0.2-mm layer on the
end surfaces as shown in the inset. The Meissner
hole is represented by a wire carrying excess current i e 50.73 A at z 0 55.2 mm below the surface.
~g! A better fit of sharp extrema ~marked by arrows! in the center is obtained for three-wire
simulation of the Meissner hole (i e 50.1 A, i s
50.27 A, z 0 53.5 mm, and d x53.5 mm! corresponding to increased current at the top and bottom of the hole.
5626
V. K. VLASKO-VLASOV et al.
56
FIG. 5. Images of the Meissner hole for H i c on the ac and ab faces after polishing the sample in Fig. 4 down to 27 mm. ~a! Trapped
flux (H50) after application of 21.5 kOe, ~b! 178 Oe, and ~c! 1171 Oe after ~a!, T550 K. During polishing, a part of the sample was
broken away. ~d! and ~e! Observation on the left end face at 50 K and H50 and H5178 Oe, respectively.
Observations on the ab plane of thicker crystals show
smoother images of the remagnetization front than those in
thin samples @compare Figs. 4~e! and 1~d!#, which can be
associated with the larger distance of the Meissner holes located close to the middle plane from the surface. This is
confirmed by observations during successive polishing of the
sample with initial thickness of 110 mm. In the thick crystal
the remagnetization front looks wide @Figs. 4~d! and 4~e!#.
But it becomes more narrow with thinning the sample. Pictures for the final thickness of 27 mm are presented in Fig. 5.
An anomalous vortex concentration at the front appears
when the thickness becomes less than ;40 mm and movable
flux bundles ~macroturbulent structures2! appear on the front
sides. Moving macrovortices were present even in thicker
crystals, but their images were smeared. Observations at the
end faces of the samples showed sharp and wavy boundaries with a flux concentration on the sides ~appearing at the
beginning of remagnetization and disappearing with increasing negative field!, independently of the sample thickness
@see Figs. 4~b! and 5~c!#.
Flux profiles measured across the ab face of the 27-mmthick sample in successively increasing positive fields after
magnetization in the negative H of 21.5 kG are shown in
Fig. 6~a!. To fit the profile at 256 G showing the sharpest
features, we calculated B n (x) above the surface of an infinitely long plate of the same thickness and width as the
sample. Positive and negative critical currents J c 5const
were assumed to flow in two halves of the plate with additional lines of ‘‘excess current’’ i ei at some distances from
the edges and at some depth from the sample surface to
represent the Meissner holes @see insets in Fig. 6~b!#. Sharp
changes of B n (x) near sample edges determined by surface
Meissner currents were simulated by homogeneous currents
of the density J M flowing in the l;0.2 mm layer on the end
faces. Meissner currents on the ab plane increasing towards
the edges should not change the results essentially. In part,
their effect is accounted by the value of J M . Values of J c ,
J M , and i ei and positions of the excess current lines were
fitting parameters. Fitting curves reasonably reproducing flux
profiles measured at 50 K @Fig. 6~c!# give ^ J c & ;
1.283105 A/cm2 ,
^ i ei & ;1.5 A, and ^ J M & ;1.12
3107 A/cm2 ~exact values are shown in the figure captions!.
If J M ;(B c1 / m 0 )/l, then B c1 ;280 G, and in the isotropic
approximation we should have i ei 5i M h 5 p (B c1 / m 0 ) 2 /J c
51.23 A, which is ;0.81 of the fitting value and could be
considered in good agreement with the experiment. However, as shown below for anisotropic materials, this correspondence turns out to be fortuitous. Additional contributions are necessary for an explanation of the excess current
along the Meissner hole in high-temperature superconducting ~HTSC! crystals.
An important disagreement follows from comparison of
the fitting parameters for the Meissner hole position and an
56
MEISSNER HOLES IN SUPERCONDUCTORS
estimate of the Meissner hole diameter D M . Using the above
simple model, D M 52(B c1 / m 0 )/J c ;35 mm, which exceeds
the sample thickness and cannot be afforded by positions of
wires fitting centers of Meissner holes in Fig. 6~b! ~18 and
9.5 mm below the top plane of the sample!. The observed
bending of the Meissner hole along the sample thickness
~Fig. 5, edge! implies that its real diameter should be less
than ;0.5 of estimated D M .
5627
D. Model of the Meissner hole in an anisotropic superconductor
So the isotropic approximation gives a surpassing value of
D M . But quite reasonable Meissner hole dimensions are obtained using the model accounting for the anisotropy of the
vortex energy and the pinning force. It results in an elliptical
shape of the Meissner cylinder and brings essential complications in the problem. However, it is easy to estimate values
and the ratio of radii for the vortex loop segments aligned
mostly with the ab plane and those along the c axis. One
FIG. 6. ~a! Profiles of B n (x) measured at the
ab face of the sample in Fig. 5 at 50 K ~vertical
scans!. At 256 Oe profiles along two different
tracks are presented. ~b! and ~c! Fits ~dashed
lines! of the profile at 256 Oe for one- and threewire representations of Meissner holes. Calculations are made at 2 mm above a 327-mm-wide
and 27-mm-thick infinite plate with currents flowing as shown in the insets. A noncentral position
of the minimum on the profile due to the sample
shape and inhomogeneity is accounted for by allowing slightly different values of J c and J M in
two halves of the plate. Fitting values in both ~b!
and ~c! are J c1 51.223105 A/cm2 , J c2
51.353105 A/cm2 ,
J M 1 51.293107 A/cm2 ,
J M 2 59.53106 A/cm2 , (x 1 ,z 1 )5(123 mm, 18
mm!, x 2 5222 mm. For one-wire simulation
of Meissner holes ~b! i e1 51.43 A, i e2 51.6 A,
and z 2 59.5 mm. For three-wire simulation
~c! i e1 50.45 A, and i 1s 50.45 A, i e2 50.48 A,
i 2s 50.45 A, z 2 56 mm, and d x57 mm. The total
current i ei 12i is changes a bit, but sharp extrema
due to Meissner holes ~marked by arrows! are
better reproduced.
5628
V. K. VLASKO-VLASOV et al.
56
FIG. 6. ~Continued).
should take the anisotropic line tension19 s 5«( u )
1d 2 «/d u 2 ~with u an angle between a vortex and the c axis!
instead of simply « and appropriate pinning values. With
«~u! from Ref. 20, the line tension of vortices along
the c axis is s c 5« 0 @ 11lnk21)/G2] @here « 0 5
(1/4p m 0 )(F 0 /l ab ) 2 5(1/m 0 )F 0 B cc1 /lnk with lnk;3.9 for
. The first and
YBCO# and the pinning force F p 5F 0 J ab,c
c
second superscripts in J i,c j correspond to current and induction directions, respectively. If to take the anisotropy paramequal to the
eter G5l c /l ab ;5 ~Ref. 20! and B cc1 and J ab,c
c
above fitting values of B c1 and J c , the radius for c segments
of the loop will be R c 51.12« 0 /F p 55 mm. For ab segments
s ab 5« 0 G @ (ln(kG)2111/G 2 # and the pinning force can be
estimated from the ratio J ab,ab
/J ab,c
;4.21 This gives R ab
c
c
;5R c . Accounting for such an ellipticity, we fitted B n profiles simulating Meissner holes by three wires arranged at
some distances along the ab direction as shown in Fig. 6~c!.
This fit gives the total current in three wires close to i ei in the
one-wire fit, but reproduces better sharp features of B n (x) at
the Meissner holes @compare Figs. 6~b! and 6~c!#.
The current along the surface of the elliptical Meissner
hole can be estimated by assuming that each vortex loop has
top and bottom segments of the length ( p /2)R ab along the
c
ab plane with the current density B ab
c1 / m 0 ;B c1 /G m 0 across a
unit length of the segment, and right and left segments of the
length ( p /2)R c along the c axis with the line current density
B cc1 / m 0 . With these assumptions,
c
c
i M ; ~ p / m 0 !~ R ab B ab
c1 1R c B c1 ! ; ~ 2 p / m 0 ! R c B c1
52 p ~ 1.12/lnk !~ B cc1 / m 0 ! 2 /J ab,c
;0.7 A.
c
The loss of the critical current inside the hole should be i c
5 p R ab R c J ab,c
;0.72i M . So the total current will be i M
c
2i c 50.28i M 50.2 A. This is by an order of magnitude less
than the fitting value.
For calculation of the total current due to the Meissner
hole, it is necessary to account also for an increase of critical
currents near the top and bottom of the cylinder where vortices are turned towards the ab direction. For the ratio of
J ab,ab
/J ab,c
;4 and above i M 2i c , one can get fitting value
c
c
i e2 ;1.5 A for the total current if the area of regions at the
top and bottom of the Meissner cylinder where J ab,ab
flows
c
will be S;0.42p R ab R c each. Then the total current i
50.25i M 12S(J ab,ab
2J ab,c
) flowing along the Meissner
c
c
hole will be equal to i e2 . Here the second term corresponds
to the increased current as compared to the neighboring areas
with c-oriented vortices.
An account of the increased currents near the top and
bottom of the Meissner hole explains the large effective dimensions of its image on the end face. Note that for better
fitting of the B n (x) perturbation due to the Meissner hole in
Fig. 4~f! is was necessary to add satellites i 2 as shown in the
scheme. They correspond to larger J c in the regions of
ab-oriented vortices. One-wire simulations cannot fit sharp
extrema of B n (x) at the Meissner hole @compare Figs. 4~f!
and 4~g!#.
56
MEISSNER HOLES IN SUPERCONDUCTORS
5629
A specific mechanism of remagnetization follows from
this model. Superconducting plates should remagnetize in
the perpendicular field due to asymmetric squeezing and collapse of vortex loops forming Meissner holes. The inner loop
is pressed from the side facing the sample edge by entering
vortices. Then the equilibrium of the line tension and pinning
is disturbed, the loop radius becomes smaller than R M , and
the loop collapses. The next loop, which becomes the inner
one, is closer to the sample center. It is squeezed from the
edge side, collapses, and so on. Such a process should give
different losses as compared to annihilation of a straight vortex and antivortex describing remagnetization in the usual
Bean model.
E. Flux instabilities due to increased current
in the Meissner hole
FIG. 7. Flux distribution in the crystal after application of the ac
magnetic field. Domains of opposite vortex polarities are generated
and drift slowly in the presence of the ac field. The structure freezes
at switching off the field.
The peculiar current distribution at the remagnetization
front discussed above does not only modify flux gradients,
but can also be also responsible for bending of the front and
the appearance of vortex bundles ~macroscopic current
loops2! near it. One can see an analogy with a ferromagnetic
plate where a straight domain boundary separating up- and
down-magnetized regions is known to be subject to the bending instability.23 This is the same type of instability as causes
current pinches in plasmas.24 The physical reason for the
instability is the magnetic field concentration inside the bend
of a current line which produces the pressure on the line,
stimulating its further curving.
In our case the remagnetization front can be subject to
such a bending instability due to the excess current flowing
along the front and assist formation of macroscopic current
loops. Such a scenario can be an alternative for the explanation of macroturbulent current structures suggested in Ref.
25 for anisotropic superconductors with nonlinear flux dynamics. Formation of these structures at the remagnetization
front can be an important source of losses in superconductors. As our experiments show, macrovortices are continuously generated under the action of ac fields and drift slowly
from the front inside the sample. They produce a specific
domain structure ~Fig. 7! changing during a time much
longer than the external field period. In this case each domain is surrounded by the Meissner torus formed by the
closed Meissner cylinder. This effect should be accounted
for in the analysis of the flux dynamics at high enough temperatures.
The Russian authors are grateful to the International Science Foundation ~Grant Nos. RF100 and RF1300! and the
Russian National Program for High-T c superconductivity
~Project No. 93211!. The work was supported by the NISIndustrial Partnering Program and by the U.S. DOE, BESMaterials Sciences, under Contract No. W-31-109-ENG-38.
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1
2
III. CONCLUSIONS
In summary, the present paper shows that remagnetization
of flat superconductors in normal fields takes place by asymmetric squeezing and collapse of closed vortex loops centered at some line in the sample rather than by motion of a
2D remagnetization front. As will be shown in forthcoming
work,22 similar behavior is observed even for plates in parallel fields and seems to be universal for any finite superconductor except films with a thickness below l. As a result,
flux-free cylinders surrounded by vortex loops are formed
along the lines of changing flux polarity. An excess current
due to special features of the cylinder surface separating
Meissner and vortex phases and increased critical current in
the shell coating the cylinder ~in regions of ab-oriented segments of vortices! give an essential local increase of the current along the Meissner hole. This can cause dynamical flux
instabilities at the front similar to the pinch effect instability
in plasmas.
ACKNOWLEDGMENTS
7
8
5630
13
V. K. VLASKO-VLASOV et al.
Y. Yeshurun, A. P. Malozemoff, F. Holtzberg, and T. R. Dinger,
Phys. Rev. B 38, 11 828 ~1988!.
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