INSTITUTE OF PHYSICS PUBLISHING
REPORTS ON PROGRESS IN PHYSICS
Rep. Prog. Phys. 65 (2002) 651–788
PII: S0034-4885(02)15613-2
Magneto-optical studies of current distributions in
high-Tc superconductors
Ch Jooss1 , J Albrecht2 , H Kuhn2 , S Leonhardt3 and H Kronmüller2
1
2
3
Institut für Materialphysik, University of Göttingen, D-37073 Göttingen, Germany
Max-Planck-Institut für Metallforschung, D-70569 Stuttgart, Germany
Max-Planck-Institut für Festkörperforschung, D-70569 Stuttgart, Germany
Received 5 January 2001, in final form 11 December 2001
Published 10 April 2002
Online at stacks.iop.org/RoPP/65/651
Abstract
In the past few years magneto-optical flux imaging (MOI) has come to take an increasing role
in the investigation and understanding of critical current densities in high-Tc superconductors
(HTS). This has been related to the significant progress in quantitative high-resolution magnetooptical imaging of flux distributions together with the model-independent determination of the
corresponding current distributions. We review in this article the magneto-optical imaging
technique and experiments on thin films, single crystals, polycrystalline bulk ceramics, tapes
and melt-textured HTS materials and analyse systematically the properties determining the
spatial distribution and the magnitude of the supercurrents. First of all, the current distribution
is determined by the sample geometry. Due to the boundary conditions at the sample borders,
the current distribution in samples of arbitrary shape splits up into domains of nearly uniform
parallel current flow which are separated by current domain boundaries, where the current
streamlines are sharply bent. Qualitatively, the current pattern is described by the Bean model;
however, changes due to a spatially dependent electric field distribution which is induced by
flux creep or flux flow have to be taken into account. For small magnetic fields, the Meissner
phase coexists with pinned vortex phases and the geometry-dependent Meissner screening
currents contribute to the observed current patterns. The influence of additional factors on
the current domain patterns are systematically analysed: local magnetic field dependence of
jc (B), current anisotropy, inhomogeneities and local transport properties of grain boundaries.
We then continue to an overview of the current distribution and current-limiting factors of
materials, relevant to technical applications like melt-textured samples, coated conductors and
tapes. Finally, a selection of magneto-optical experiments which give direct insight into vortex
pinning and depinning mechanisms are reviewed.
(Some figures in this article are in colour only in the electronic version)
0034-4885/02/050651+138$90.00
© 2002 IOP Publishing Ltd
Printed in the UK
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Ch Jooss et al
Contents
1. Introduction
2. The critical state
2.1. Bean’s critical state model
2.2. Microscopic understanding
2.3. The mesoscopic level
2.4. Flux dynamics
2.5. Thermally activated decay
3. Flux imaging using magneto-optics
3.1. Faraday effect
3.2. Experimental set-up and resolution
3.3. Calibration of the flux density
3.4. Determination of supercurrents
4. Current distribution and sample geometry
4.1. Thin strips
4.2. Arbitrarily shaped films—critical state
4.3. Thin films—partly penetrated state
4.4. Finite electric fields
4.5. Convex corners, crooked edges and holes
4.6. Superconductors of finite thickness
5. Meissner currents and surface barriers
5.1. Meissner expulsion
5.2. Modification of the critical state
5.3. Macroturbulence and instabilities
5.4. Surface barriers
5.5. Reversible properties in thin films
6. Anisotropic and inhomogeneous currents
6.1. Anisotropic critical currents
6.2. Field-dependent critical currents
6.3. Vortex phase transitions
6.4. Domain boundaries with perpendicular currents
6.5. Domain boundaries with parallel currents: I
6.6. Domain boundaries with parallel currents: II
7. Texture and local current distribution
7.1. Grain boundaries
7.2. Polycrystals
7.3. Melt-textured material
7.4. YBCO-coated conductors
7.5. Bi-2212 and Bi-2223 tapes
8. Pinning and depinning mechanisms
8.1. Columnar defects
8.2. Pinning mechanism at antiphase boundaries in thin films
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Current distributions in high-Tc superconductors
8.3. Surface pinning effects
9. Summary and conclusions
Acknowledgments
References
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Ch Jooss et al
Index of symbols
If a quantity is used only once it may not appear in this index. Some symbols have several
meanings which depend on the context.
α
αj
0
κ
φf whm
0
λ
λab , λc
µ0
ρ
ρd
θ
θf whm
τ
ξ
ξ0
ξn
ψ
a, aF LL
a
Aj
b
B
Ḃ
B̃
Bs
Bff
CC
d
E
FH
fp
Fp
FL
Fη
g
G
GB
Faraday rotation angle. Angle of current discontinuity lines
Angle of current streamlines with respect to a planar defect
Gap function
Vortex line energy per unit length
Ginzburg–Landau parameter
Full width at half-maximum of the in-plane texture
Magnetic flux quanta
Anisotropy ratio = λ/λab
London penetration depth of a magnetic field. Light wavelength
Anisotropic London penetration depth in the (a, b) and in the
c-direction, respectively
Magnetic permeability in space
Electric resistivity
Electric sheet resistivity ρd = ρ/d
Grain boundary tilt angle. Current orientation at current domain
boundaries
Full width at half-maximum of the out-of-plane texture
Thickness of the current layer seen by flux imaging of thick
samples
Ginzburg–Landau coherence length
BCS coherence length
Proximity length of the condensate in a normal-conducting area
Order parameter of the superconducting condensate
Lattice constant of the flux line lattice
Half of the sample width a = W/2
Anisotropy ratio of current densities
Microscopic magnetic flux density. Half of the sample’s width
in rectangular samples
Mesoscopic magnetic flux density
Time derivative of the magnetic flux density
Fourier coefficients of B
Step in the flux density due to current strings at the sample’s
surface.
Step in the flux density due to a current string at the flux front.
Coated Conductor
Sample thickness
Mesoscopic electric field
Mesoscopic Hall force density per unit volume
Microscopic elementary pinning force
Mesoscopic pinning force density per unit volume
Mesoscopic Lorentz force density per unit volume
Mesoscopic friction force density per unit volume
Local magnetic moment, current potential function
Local sheet moment G = gd
Grain boundary
Current distributions in high-Tc superconductors
h
H
Hc1
Hc2
Hex
Hk
Hp
HAGB
I
I
I0
I1
IBAD
j
J
j0
j 1 , j2
J1 , J2
jb
jc
jc,ab
jc,c
jc,L
jc,T
jL
jT
Jc
jM
jv
jv K0 , K1
Kg
l
le
ls
LAGB
LD
meq
M
Ms
n
n
Measurement height = distance between top surface of the
superconductor and the MOL
Mesoscopic magnetic field
First critical field of a type-II superconductor
Second critical field of a type-II superconductor
External magnetic field
Anisotropy field of the MOL
Field of full flux penetration of a superconductor
High-angle grain boundary
Electric current. Light intensity
Light intensity reflected out of a MOL
Light intensity of the incident beam
Background light intensity due to non-ideal polarizers
Ion-beam-assisted deposition
Electric current density
Sheet current J = d j
Depairing current density of a superconductor
High and low values of the current density in a superconductor
with inhomogeneous jc
High and low values of the sheet current in a superconductor with
inhomogeneous jc
Critical current density through a GB
Critical current density
Critical current density in the a–b plane
Critical current density in the c-direction
Critical current density longitudinal to planar defects
Critical current density transverse to planar defects
Longitudinal component of a current density
Transverse component of a current density
Critical sheet current Jc = djc
Meissner current density
Microscopic eddy current density of a vortex
Mesoscopic current density due to vortex density gradients
Modified Hankel functions of zeroth and first order
Integral kernel
Length
Electronic screening length at a GB
Strain decay length at a GB
Low-angle grain boundary
Linear defect
Equilibrium magnetization normalized to Hc1
Magnetization
Spontaneous magnetization
Exponent of the non-linear current–electric field relation E ∝ j n
Normal vector
655
656
Q
ptr,T
P
R
RABiTS
t
t0
T
Tc
Tk
Tirr
U
v
V
Vc
W
Ch Jooss et al
Penetration depth of the flux front measured from the sample’s
centre. Integral kernel
Transverse component of the transport scattering probability
tensor
Penetration depth of the flux front measured from the sample’s
edge
Sample radius
Rolling-assisted biaxially textured substrates
Time
Time decay constant to reach a nearly stationary state
Temperature
Critical temperature
Temperature of compensation in ferrimagnets
Irreversibility temperature
Activation barrier for flux creep
Velocity field of moving flux B
Volume segment, sample volume, Verdet’s constant
Correlation volume, flux bundle volume
Sample’s width. Transmission amplitude of Cooper pairs through
a surface
Current distributions in high-Tc superconductors
657
1. Introduction
During all phases of investigation of magnetic and current-carrying properties of
superconductors it has been considered desirable to apply a method for the measurement
of the spatial distribution of the magnetic flux density. In particular, to achieve a profound
microscopic understanding of global properties of superconductors such as magnetization,
magnetic susceptibility and critical transport currents it is necessary to use a local probe.
The magneto-optical Faraday effect provides a method which allows one to combine
relatively high spatial resolution and magnetic sensitivity with short measurement times and
large imaging areas. Due to the progress in magneto-optical measurement techniques as well as
the progressive requirements of the research into high-temperature superconductors (HTS), this
method has become used to a greater and greater extent during the last ten years. Since, up to
now, no significant magneto-optical Faraday effect has been observed in superconductors, one
has to rely on well suited magneto-optical layers (MOLs) as magnetic field sensing elements.
The first magneto-optical visualizations of magnetic flux distributions were carried out by
Alers (1957) and De Sorbo (1960) on type-I superconductors using thick discs of Ce(PO3 )3 or
Ce(NO3 )3 which enabled a spatial resolution of 200 µm. Using this method, the flux distribution of a number of metallic superconductors was investigated and presented in a review article
(De Sorbo and Healy 1964). A great improvement was achieved by Kirchner (1968) (see also
Kirchner (1969)) by using a thin-film technique based on the paramagnetic europium chalcogenides and halogenides EuS and EuF2 , leading to the so-called high-resolution magnetooptical technique. Hübener et al (1970) and Habermeier and Kronmüller (1977) applied this
technique to the measurement of flux line density gradients in type-II superconductors. Application of such MOLs to HTS was performed by Moser et al (1989) and Forkl et al (1990).
The magneto-optical imaging of flux patterns in HTS was accompanied by significant
improvements in the field of magneto-optical active materials. Using single-component EuSe
instead of EuS/EuF2 mixtures, the evaporation processes have become easier and, in addition,
EuSe exhibits a slightly larger Faraday rotation (Dutoit and Rinderer 1987, Schuster et al 1991).
However, the extremely large Faraday rotation of the Eu chalcogenides (up to 110◦ µm−1
for EuSe at 4.5 K, magnetic fields of µ0 Hex = 1.15 T and wavelength of the polarized light
λ = 560 nm (Schoenes 1975)) strongly decreases at temperatures above the antiferromagnetic–
paramagnetic transition (Tc = 4.6 K) and consequently the application to HTS is limited to a
low-temperature regime T < 15 K.
A completely different kind of magneto-optical technique has been developed by
Polyanskii et al (1989) with the application of ferrimagnetic, so-called garnet or bubble
domain films consisting of bismuth- or gallium-doped yttrium–iron garnets (YIGs) (Polyanskii
et al 1990b, 1990c, Szymczak et al 1990, Indenbom et al 1990, Gotoh et al 1990, Belyaeva
et al 1991b, Gotoh and Koshizuka 1991). These garnets are grown by liquid phase epitaxy
and exhibit magnetic ordering below Tc = 400–600 K with a spontaneous magnetization
vector normal to the film plane. Below their point of compensation Tk < Tc , large Faraday
rotations parallel to the magnetization vector are obtained (up to 4◦ µm−1 in Y3−x Bix Fe5 O12
at room temperature and λ = 426 nm (Simsa et al 1984)) which depend strongly on
the Bi content. In the temperature region of superconductivity T < 150 K, the Faraday
rotation is nearly temperature independent. These films show a labyrinth-like pattern of stripe
domains which appear as a transparent and opaque structure superimposed on the image of the
underlying superconductor. The measurement of the magnetic flux density is only indirect:
one observes the changes of the magnetic domain density during flux penetration into the
superconductor. The field-dependent size of the magnetic domains of 2–30 µm limits the
spatial resolution.
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A great step forward in the application of the magneto-optical technique was the
development of (Lu, Bi)-doped iron garnets with in-plane magnetization vectors (Wallenhorst
et al 1995, Grechishkin et al 1996, Ubizskii et al 1996). To our knowledge these planar films
were applied first to HTS by Dorosinskii et al (1991); see also (Belyaeva et al 1991a, VlaskoVlasov et al 1991, Dorosinskii et al 1992). An increasing normal field successively rotates
the magnetization vector out of the plane and thus leads to an increase of the Faraday rotation
normal to the film in the range of 7◦ –9◦ µm−1 at λ = 633 nm (Okuda et al 1990, Adachi et al
2000) in the saturated state. In contrast to the iron garnets with perpendicular anisotropy, the
planar films exhibit a direct (non-linear) relation between the local normal component of the
magnetic flux density Bz and the Faraday rotation angle, similar to paramagnetic MOLs. This
and the absence of coercivity strongly increase the magnetic sensitivity of the planar films up
to 10 µT. The spatial resolution of the field measurement is only limited by the film thickness
of the MOL and the measurement height between the MOL and the superconductor surface,
typically leading to spatial resolutions of some µm (see section 3.2 for more details). For the
measurement of the magnetic domain structure of tapes, used for magnetic recording, a spatial
resolution of 370 nm has been achieved (Grechishkin et al 1996). Recently, an achievement
which was desired for a long time, the magneto-optical imaging of individual flux lines, has
been achieved by Goa et al (2001) (see section 3.2).
The investigation of the local current density of superconductors requires not just a
qualitative visualization of the magnetic flux patterns in superconductors. For a quantitative
analysis of the current density it is necessary to calibrate the measured light intensity contrast
into a magnetic flux density distribution. First steps towards quantitative imaging of the
magnetic flux density have been achieved by Forkl et al (1991) for paramagnetic MOLs and
by Dorosinskii et al (1992) for garnets with in-plane magnetization. Complete calibration
functions for both kinds of MOL, including the non-linear transfer function of the polarization
microscope, were given by Jooss et al (1996a) and Johansen et al (1996).
A second fundamental development was the quantitative determination and understanding
of the supercurrent density distribution on the basis of the measured flux density. Since
one measures only one component of the flux density distribution B (r ) (usually the normal
component Bz ) at a plane r = (x, y, z = constant) above the superconductor, the relation
between self-field and current distribution is non-trivial and strongly depends on the sample
geometry. The anisotropic crystal lattice in a HTS assists the growth of flat samples where the
curvature of the magnetic field lines of the self-fields is strong and the original Bean model
which is valid for long cylinders fails. Surfaces and inner interfaces, e.g. grain boundaries
and cracks, inhomogeneity and anisotropy of jc give rise to complex current patterns. All this
promoted the development of a variety of models in order to relate the measured flux density
patterns to screening and critical current distributions. An alternative approach has been
developed with the progress in algorithms for numerical inversion of the Biot–Savart law which
gives an integral relation between the current density distribution and the magnetic self-field
(Wijngaarden et al 1996, Pashitskii et al 1997, Wijngaarden et al 1997, Jooss et al 1998a); for
an one-dimensional solution see Johansen et al (1996). It opened up the possibility of a modelindependent determination of supercurrent distributions by magneto-optical imaging with high
spatial resolutions up to 1 µm. This method enabled significant progress from flux to current
imaging by application of magneto-optical techniques. It allows the investigation of more
complex problems, such as inhomogeneous and anisotropic current densities, the dependence
of critical currents on the local magnetic field, time- and thus electric field-dependent current
densities and the details of current distributions at grain boundaries and other inner interfaces.
This article summarizes the magneto-optical technique for the determination of flux and
current density distribution in superconductors. It focuses mainly on the present state of
Current distributions in high-Tc superconductors
659
understanding of the spatial distribution of supercurrents in HTS and is organized as follows.
Section 2 gives a summary of the critical state theory in homogeneous bulk HTS, starting
with the original Bean model. It describes the relation between the microscopic properties
of vortex lines and pinning forces and the measured mesoscopic and macroscopic properties
of the critical state: the local and global critical current and the magnetization. Section
3 summarizes the magneto-optical measurement technique and the calibration problem. It
introduces different methods for the determination of supercurrent distributions from the
measured flux distributions. The following parts, sections 4–7, analyse systematically various
effects which are determining the spatial distribution of the supercurrent density within HTS
samples. Section 4 considers the relation between sample geometry and current distribution
in the subcritical and critical state. The splitting of the current pattern into current domains
separated by current domain boundaries is analysed and particular attention is paid to the role
of the electric field distribution. The magneto-optical observation of reversible properties of
the HTS and the modification of the critical state due to Meissner currents are considered
in section 5. The current distribution may be strongly modified due to spatially varying
pinning strengths, e.g. due to spatial variations of the film growth or partly irradiating the
superconducting samples. This has deep consequences for the current domain patterns and
may give rise to huge local electric fields. Intrinsic anisotropy due to the layered structure as
well as microstructural anisotropies are strongly influencing the current distributions. All these
effects are described in section 6 as a first step towards the understanding of complex current
patterns of technical materials addressed in section 7. Here, the influence of grain boundaries
and complex microstructures on the current distribution is revealed in polycrystalline and
partly textured HTS, such as melt-textured YBa2 Cu3 O7 , Bi2 Sr2 Ca1 Cu2 O8 or Bi2 Sr2 Ca2 Cu3 O10
monofilamentary and multifilamentary tapes and YBa2 Cu3 O7 -coated conductors. Section 8
presents a selection of recent magneto-optical studies of flux pinning at well defined and well
characterized defects in the crystal lattice and section 9 summarizes the results.
2. The critical state
Up to now the capability of carrying electric currents where the losses are reduced to an
unmeasurably small level is the most astonishing feature of superconductors. At the same time
the maximum loss-free current density jc is the most important property for their technical
application. A large loss-free total transport current I requires high supercurrent densities
in the entire volume of a superconductor. However, in thermal equilibrium the supercurrent
is only able to flow in a small surface layer in superconductors of all kinds. Generally, the
phase coherence of charge carriers in a condensed state is inconsistent with a bulk magnetic
field which is inevitably present in the volume of a superconductor with a bulk supercurrent.
This is equivalent to an ideal diamagnetic screening which is, in combination with ‘vanishing’
resistivity, usually taken as a criterion for the presence of a superconducting state.
Let us first consider the Meissner state, where an external magnetic field is completely
screened from the bulk superconductor (London 1935). The screening takes place due to
Meissner supercurrents with density jM , flowing in a surface layer of the superconductor with
thickness of the order of the London penetration depth λ. The maximum of jM is given by the
depairing current density
1
0
j0 = √
,
(1)
3 2 πµ0 ξ λ2
where the kinetic energy of supercurrents balances the condensation energy. 0 = h/2e
denotes the elementary flux quantum, λ the magnetic penetration depth and ξ the coherence
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Ch Jooss et al
Figure 1. (a) The magnetization curve of a pure, defect-free type-II superconductor. (b) A sketch
of the magnetic field lines and the Meissner screening currents jM for a cylindrical superconductor
in the Meissner state (cross section). (c) A cylindrical superconductor for H > Hc1 . The average
current of an ideal Abrikosov lattice is zero except for the superconductor surface, where the eddy
currents of the vortices jv are directed oppositely to the Meissner surface currents.
length. In the Meissner state, due to the restriction of the supercurrent to a surface layer, most
of a superconductor’s cross section does not contribute to a supercurrent-carrying state.
This is also valid for clean and defect-free type-II superconductors in the mixed state.
For magnetic fields H Hc1 , the superconductor lowers its energy by the penetration of
quantized magnetic flux. In thermal equilibrium these flux quanta (also flux lines, vortex
lines) arrange in a periodic lattice (Abrikosov 1957). On mesoscopic length scales >λ, the
superconducting eddy currents, jv , surrounding each flux line average to zero, despite there
being a surface region where the eddy currents of the vortices are directed oppositely to the
Meissner screening current and thus reduce the ideally diamagnetic screening (see figure 1).
As early as 1930 type-II superconductors were detected by Shubnikov and de Haas (1930).
After a brief flurry of experimental interest in the mid-1930s, the field lay dormant and was
revitalized at the beginning of the 1960s. The experimental observation of irreversible and
hysteretic magnetization curves together with the measurement of bulk critical currents inspired
Bean (1962) to propose a model which allows an empirical understanding of the experiments.
The Bean model just assumes that superconductors in the critical state can carry volume
supercurrents with a constant critical value jc and thus behave as permanent magnets. In
contrast to the reversible magnetization of the Meissner and Shubnikov state which is generated
by surface currents, the irreversible magnetization of the Bean critical state (which depends on
the magnetic history of the sample) is due to bulk currents and magnetic flux gradients covering
large regions of the superconductor. It was recognized that these irreversible properties are
caused by the interaction of flux lines with material defects and inhomogeneities being present
in real type-II superconductors. The critical state is in reality a metastable non-equilibrium
state in mixed-phase superconductors, where magnetic vortices are hindered from reaching
their equilibrium positions by their interactions with defects in the crystal lattice. Thermally
activated creep of the vortex structures gives rise to a weakly dissipative critical current, to a
strongly non-linear relation between electric field E and current density j and is manifested
in a time decay of magnetization currents.
2.1. Bean’s critical state model
The simplest possible critical state model for obtaining the magnitude and distribution
of currents in type-II superconductors is given by the Bean model (Bean 1962, 1964).
Current distributions in high-Tc superconductors
661
B z =<b z >
(1)
(2)
0
0
ex
ex
(2)
j y =<j v >
jc
(1)
jc
ex
(3)
(4)
(3)
(4)
0
ex
jc
0
ex
jc
Figure 2. The magnetization curve and the corresponding flux and current density distribution in
the Bean model. It assumes that the external field Hex is applied parallel to a long cylinder or slab,
a field-independent critical current jc (B) = constant and that no thermal relaxation takes place
jc (t) = constant. Reversible properties are neglected.
In its original version it simply assumes the existence of a constant critical current density
jc in that regions of a superconductor where magnetic flux has penetrated and formed a flux
density gradient. In addition, in its original version, it assumes zero current density in the fluxfree regions of the sample and consequently disregards demagnetizing effects and curvatures
of the magnetic field lines in samples of finite sizes.
For long cylinders or slabs in a parallel magnetic field Hex applied along the z-axis after
zero-field cooling (ZFC), the Bean model predicts a piecewise-linear slope |∂Bz /∂r| = µ0 jc
in that region where magnetic flux has penetrated. Note that this is only valid for regions inside
the superconductor which are far enough from the surfaces perpendicular to the external field
(see figure 2). According to the Bean model, the critical current density is a measure of the
volume pinning force density
Fp = jc Bz
(2)
exerted on the magnetic flux by defects in the crystal lattice of the superconductor. In flux-free
regions B = 0 or for a field-cooled experiment, where no flux gradients are present, one
has j = 0. The detailed pattern of B depends on the magnetic history. Figure 2 shows the
distribution of magnetic field and critical current density when Hex is increasing from zero to
a maximum value and afterwards is decreasing to zero again.
A characteristic quantity of the Bean model is the field Hp at which in a gradually
increasing applied field Hex the magnetic flux has penetrated to the centre of the sample and
the supercurrent density has reached its saturation (critical) value jc over the entire specimen.
For a long cylinder with thickness d R (R denotes the radius) and for strips and slabs in
the parallel limit (d W ) one has
Hp =
Wjc
,
2
(3)
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Ch Jooss et al
a)
b)
f
bz
Figure 3. Visualization of the properties of an individual Abrikosov flux line in YBCO parallel to
c-axis (λab = 150 nm, ξab = 1.5 nm). The axial flux density bz (a) and the absolute value of the
order parameter |ψ| (b) are shown as a surface plot. This approximate Ginzburg–Landau solution
was given by Clem (1975).
where W is the lateral extent of the specimen. The penetration field Hp may be used to estimate
the critical current density from space-resolved measurements of the flux penetration.
Note that the Bean model totally disregards thermal relaxation effects and replaces the
real non-linear E(j ) curve of a superconductor by a stepwise dependence E = 0 for j jc
and E → ∞ for j > jc . Further, in its simplest form presented above, the magnetic field
dependence of jc as well as the reversible properties are neglected; that is, B(H ) = µ0 H ,
which is valid only for Hc1 = 0.
2.2. Microscopic understanding
Even if Bean (1962) recognized that the microstructure of a superconductor is determining
the critical state, the microscopic origin—an interaction between magnetic flux quanta and
crystal defects—was not fully known about at the time of his first paper. Since the presence
of bulk magnetic flux is inconsistent with the phase coherence of paired electrons in the
superconducting state, the Bean proposal required the assumption of a microscopically
inhomogeneous superconducting state. He and others (see for example Gorter (1962a, 1962b),
Bean and Doyle (1962)) discussed the Mendelssohn mesh or sponge model (Mendelssohn
1935), picturing such a material as a fine filamentary mesh of superconducting and nonsuperconducting phases with a length scale of spatial variation smaller than the magnetic
penetration depth of a bulk superconductor. In his review, two years later, Bean (1964) took
the observation of flux quantization into account, recognizing that the main aspects of his
phenomenological theory do not depend on the microscopic details.
For external fields Hc1 Hex Hc2 , the magnetic field enters type-II superconductors in
the form of magnetic flux quanta, each carrying a single flux quantum 0 = h/(2e). In pure
superconductors the flux lines arrange in a periodic lattice. This two-dimensional lattice of flux
quanta was predicted by Abrikosov (1957). Flux quantization was proven experimentally by
Deaver and Fairbank (1961) and Doll and Näbauer (1961) independently. The arrangement of
quantized flux lines in a periodic lattice was experimentally observed by the neutron scattering
(Cribier et al 1964) and the Bitter decoration technique (Träuble and Essmann 1968).
Figure 3 shows the axial magnetic flux density distribution bz (r) (r = (x 2 + y 2 )1/2 ) and
the spatial variation f (r) of the order parameter ψ(r) = f (r)eiϕ for a vortex line in a bulk
superconductor using an approximate Ginzburg–Landau solution (Clem 1975). In the Clem
model the order parameter suppression at the vortex core is given by
r
f (r) = ,
(4)
2
r + ξv2
Current distributions in high-Tc superconductors
663
j y = <jv >
jc
Figure 4. Generation of a bulk supercurrent density jy = jv due to positive and negative density
gradients of microscopic flux quanta in the x-direction.
√
2ξab . The superconducting eddy currents
2
r + ξv2
r
0
jv =
K
1
3
3
2
2πµ0 λab
λab
r + ξv2
with core radius ξv =
screen the magnetic flux density
0
bz =
K0
2πµ0 λ2ab
r 2 + ξv2
λ3ab
(5)
(6)
of the vortex from the bulk superconductor. The order parameter is suppressed at the centre of
the vortex, where the flux density reaches a maximum, leading to a ‘normal-conducting’ core
of the vortex. If flux lines are distributed homogeneously, the averaged supercurrent density
j vanishes in the bulk of the superconductor. The application of a transport current in the
vortex state of a superconductor requires an inhomogeneous flux line distribution since the
macroscopic current is then mediated between the eddy currents of the single flux lines. In this
case, the microscopically closed current loops superimpose to a macroscopic flowing current
density (see figure 4).
Also for the HTS, the existence of flux lines and the Abrikosov lattice has been proven
experimentally by different techniques. Vortices have been observed by Bitter decoration
technique (Gammel et al 1987, Vinnikov et al 1988, Dolan et al 1989, Bolle et al 1991,
Grigorieva et al 1993, Grigorieva 1994, Dai et al 1994, Yao et al 1994), neutron scattering
study (Forgan et al 1990, Yethiraj et al 1993, Cubitt et al 1993, Keimer et al 1994a, 1994b),
scanning tunnelling microscopy (Maggio-Aprile et al 1995, Renner et al 1998) and magnetic
force microscopy (Hug et al 1994, Moser et al 1995).
Comparing the vortex lines in HTS with the vortex lines in formerly known conventional
superconductors, one finds the following main new features:
(i) Due to the layered structure of the HTS a flux line extending perpendicular to the CuO2
plane cannot be considered as a homogeneous line. In the HTS with the largest anisotropy
ratio = λc /λab , Bi2 Sr2 Ca1 Cu2 O8 ( = 150), a vortex line consists of a stack of nearly
2D ‘pancake’ vortices with Josephson coupling between the layers (Bulaevskii 1990, Clem
1991, Artemenko and Latyshev 1992, Feinberg 1992, 1994). In YBa2 Cu3 O7 ( = 5), the
layered structure of the vortex line is less pronounced and a flux line resembles more an
Abrikosov vortex.
(ii) The layered structure together with the larger thermal phase fluctuations of the condensate
in HTS give rise to new collective properties of the vortices; in particular, new vortex
phases and a first-order melting transition are observed (Zeldov et al 1995, Revaz et al
664
Ch Jooss et al
1996, Liang et al 1996, Welp et al 1996, Roulin et al 1996, Crabtree and Nelson 1997,
Crabtree et al 1998).
(iii) The structure of flux lines is influenced by the symmetry of the order parameter. For a
pure dx 2 −y 2 -wave superconductor one expects a fourfold symmetry of the order parameter
near the vortex core (Ren et al 1995, Berlinsky et al 1995, Franz et al 1996b, Ichioka
et al 1996, Wang and Wang 1996). However, experiments on YBa2 Cu3 O7 show elliptical
vortex cores (Maggio-Aprile et al 1995), possibly related to the orthorhombic distortion
of the crystal lattice in this material.
(iv) Theory suggests that the vortex lattice in a dx 2 −y 2 superconductor exhibits different
symmetries—square or hexagonal lattices—depending on temperature and field (Maki
and Won 1996). In YBa2 Cu3 O7 , a hexagonal (Gammel et al 1987), a square (Yethiraj et al
1993) and some mixed symmetry (Keimer et al 1994a, 1994b, Maggio-Aprile et al 1995)
of the vortex lattice have been observed. Small-angle neutron diffraction investigation of
Bi2 Sr2 Ca1 Cu2 O8 (Cubitt et al 1993) shows a hexagonal symmetry of the vortex lattice.
(v) In conventional superconductors, the ‘normal’ vortex core is characterized by a continuum
of bound states lying in the gap region of the quasiparticle spectrum. In contrast, the
quasiparticle density of states in the vortex core in YBa2 Cu3 O7 and Bi2 Sr2 Ca1 Cu2 O8
is almost empty (Maggio-Aprile et al 1995, Renner et al 1998). Possibly, this implies
consequences for the dissipation mechanism during vortex motion and for the vortex
pinning mechanisms at lattice defects.
Recently, the coexistence of ferromagnetism and superconductivity in the same regions was
directly observed by magneto-optical imaging in the RuSr2 (Gd0.7 Ce0.3 )2 Cu2 O10 compound on
length scales of 10 µm (Chen et al 2001), raising the question of the nature of the flux lines in
this system.
2.3. The mesoscopic level
Magneto-optical measurements provide experimental insight into the critical state of
superconductors on a mesoscopic level. Only recently was the observation of individual flux
lines by magneto-optical imaging achieved, by Goa et al (2001). However, usually magnetooptical observation allows one to get experimental information about the flux and current
distribution on mesoscopic length scales 1 µm. This is the length scale which is related
to the collective behaviour of flux lines and consequently to the formation of distributions of
critical currents in HTS. In order to define the properties clearly, three different length scales
are distinguished in the following:
(i) The microscopic level of the properties of single flux lines and the physics of motion and
interaction of individual flux lines with defects.
(ii) The mesoscopic level of collective behaviour of flux lines with averaged forces, flux and
current density distributions. It is defined on length scales larger than the correlation
length Lc of collective behaviour of vortices.
(iii) The macroscopic level of the averaged transport properties of the entire superconductor.
In particular, in HTS the mesoscopic level is extremely important: it represents the length
scale on which a real transport current is created. In particular, it determines the characteristic
size of all variations in the current density distribution, e.g. the minimum size of current bends
and current domain walls, separating domains of uniform flowing currents (see section 4). The
phenomena of non-linear current flow due to thermal relaxation processes with characteristic
local E(j ) curves mainly manifests on this level. The material properties of ceramic HTS
Current distributions in high-Tc superconductors
a)
665
b)
c)
Figure 5. Visualization of new aspects of vortices in HTS compared to conventional
superconductors. (a) The limit of Abrikosov and pancake vortices. (b) A vortex core in a dwave superconductor with induced s-wave components (Wang 1996). (c) A sketch of quasiparticle
bound states in the vortex cores of conventional (BCS-like) superconductors and HTS.
with a rich variety of inhomogeneities and defects often lead to highly non-uniform current
distributions.
In the framework of the Bean model, the relation between levels (i) and (ii) is given by
statistical averaging of the elementary pinning force fp , the microscopic current density jv
and flux distribution b of the single vortices
fp a
,
Fp =
j = jv a ,
B = ba .
(7)
V
The averaging is performed on a length scale of some lattice parameters a of the flux line lattice;
V denotes the averaged volume segment. The macroscopic volume pinning force Fp Volume ,
critical current density jc Volume and the magnetization M represent averaged properties of the
whole sample.
The problem of the relation between the microscopic properties of vortices and single
vortex pinning forces and the mesoscopic properties is very complex. Often, many pinning
sites act collectively on single flux lines. When a vortex is displaced from its ideal lattice
position, elastic counter-forces try to move it back. Conclusively, the mesoscopic volume
pinning force density Fp in a superconductor is a result of the collective action of many
pinning sites together with the elastic forces, both on the microscopic level. First theoretical
results for the statistical summation of the microscopic forces acting on the flux line lattice
have been obtained by Larkin and Ovchinnikov (1973, 1979) and Schmucker and Kronmüller
(1974).
It is beyond the scope of this paper to review statistical summation. For the purpose of an
introduction, we discuss here two simple cases: for correlated pinning of vortices, for example
at a periodic array of line or planar defects, one has
Fp = nd fp .
(8)
This is valid as long as the vortex density is lower than the density of defects nd . The simplest
model for a statistical distribution of point defects ascribes the mesoscopic pinning force
density to the mean square variation of the microscopic pinning forces:
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Ch Jooss et al
fp 2 nd Vc
Fp =
.
Vc
(9)
Whereas the long-range order of the vortices is lost, within a bundle volume Vc ≈ Rc2 Lc one
has still short-range order. Rc and Lc are the vortex correlation lengths perpendicular and
parallel to the vortex line, respectively. There are different regimes of pinning: single vortex,
small bundles and large bundles. The limit Rc ≈ a corresponds to a so-called vortex glass
state, resembling an amorphous state of solids. The anisotropy of the HTS and the highertemperature range of superconductivity assists the destruction of long-range order by kink
deformation of vortex lines and thus the formation of a glassy state (Fisher 1989). For higher
temperatures and fields, one observes the melting of the vortex solid and the occurrence of a
vortex liquid state due to the strong fluctuating thermal forces. For reviews of the statistical
summation, see Blatter et al (1994) and Brandt (1995a).
2.4. Flux dynamics
For an analysis of the distributions of both transport and magnetization currents together with
their corresponding magnetic flux line distributions, different forces acting on the flux lines
have to be considered. If an external current is transferred through the volume of a type-II
superconductor by density gradients in the vortex system, the vortex lines start to move under
the action of a force. In the case of materials with large Ginzburg–Landau parameter κ, this
force is modelled approximately as a Lorentz force density fL of the mesoscopic current
density on the microscopic flux quanta (Anderson 1962); for a derivation see the work of
Friedel et al (1963). To avoid dissipation and a finite resistivity ρ which is due to flux motion,
the driving force on the vortices has to be counteracted by pinning forces fp of material
defects on the flux lines. A similar consideration is apposite when applying an external field
H > Hc1 to superconductors: almost instantaneously, flux lines will be formed at the sample
borders and will be driven to their equilibrium positions by mesoscopic currents created by
the inhomogeneous vortex distribution in the non-equilibrium state. If the driving force on
the vortices is counteracted by pinning forces, density gradients of flux lines and bulk critical
current are generated, forming a metastable state.
On the mesoscopic level, the volume Lorentz force density is given by FL = j × B (Gorter
1962, Friedel et al 1963, Josephson 1966). Within a perfectly homogeneous superconductor,
FL is counteracted by the friction force density Fη = −ηv , where v is the velocity field of
the moving flux. The friction coefficient is obtained by an analysis of different dissipative
processes of the moving vortices (Bardeen and Stephen 1965). The moving flux generates an
electric field E = B × v (or ∇ × E = −Ḃ ) which is related to a finite resistivity ρ.
An additional force is acting on a moving vortex in the direction parallel to the flowing
supercurrent j . Since this effect is similar to the Hall effect in an ordinary metal, this force is
called Hall force density FH = ns f v × B (Hübener et al 1979); for microscopic mechanisms
see also (Bardeen and Stephen 1965, Nozières and Vinen 1966). Here, ns is the density
of superconducting charge carriers and f denotes a dimensionless factor which depends on
the quasiparticle scattering rate. Fluctuating thermal forces δ F (T , B, t) which depend on
temperature and vortex density may lead to strong fluctuations of the vortex positions, and
give rise to a non-linear diffusion of vortices and vortex bundles and a corresponding time
decay of the flowing current density.
Summarizing all forces, the flux motion has to satisfy the following equation of motion:
j × B − ηv + ns f v × B + Fp (B ) + δ F (T , B, t) = 0.
(10)
Current distributions in high-Tc superconductors
667
In the superclean limit l ξ , l denoting the quasiparticle mean free path, the flux motion
in type-II superconductors is dominated by the Lorentz and Hall forces (Blatter et al 1994).
Consequently, the small coherence length in the HTS should support the observation of the Hall
force. However, up to now, the Hall effect has been observed in HTS only in the flux flow regime
near Tc (Ong 1990, Hagen et al 1993); for a short review see the article of Brandt (1995a).
At low temperatures, the Hall force is small compared to the Lorentz and pinning forces
and has not been observed by means of magneto-optics up to now. At low T or for j jc the
fluctuating force δF (T , B, t) becomes small and the pinning forces allow the formation of a
(nearly) static flux density gradient (critical state) where the driving force density is balanced
by the pinning force density (Anderson 1962, Friedel et al 1963):
jc × B + Fp (B ) = 0.
(11)
For Fp (B ) = constant we recover again equation (2) of the Bean model.
2.5. Thermally activated decay
From the thermodynamic point of view, the stationary critical state defined by equation (11)
is metastable and thus has a tendency to decay either by thermal activation or by tunnelling
processes usually called ‘flux creep’. Due to the large Ginzburg parameter, the extended
temperature regime and the vortices which are modified by the layered crystal structure of
HTS, thermal fluctuations are much stronger in HTS compared to low-Tc superconductors
(Blatter et al 1994). Consequently, the underlying concept of critical state has to be changed,
e.g. the balance of forces in equation (11) is in fact time dependent. Therefore, the Bean
critical current density has to be replaced by a critical current density which is defined by the
timescale or—what is equivalent—by the electric field or voltage level of the experiment. It
characterizes the transition between the flux creep and flux flow regimes. Another metastable
regime of thermally assisted flux flow exists at high temperatures and small driving forces (Kes
et al 1989, van den Berg 1989).
In magneto-optical experiments, usually a slowly relaxing magnetization state is observed
which is created by applying external magnetic fields Hex with a finite sweep rate. The
timescale t0 which is necessary to reach a nearly stationary (slowly relaxing) state after suddenly
switching on a magnetic field was estimated by Brandt (1993) to be
t0 (x, y) ≈
µ0 W 2
,
2πρ(x, y)
(12)
where W is the lateral size of the sample, ρ(x, y) = E(x, y)/j (x, y) is the spatially dependent
flux creep or flux flow resistivity and
E(j ) = Ec e−U (j,B,T )/kB T ,
(13)
is the non-uniform electric field in the superconductor which depends on the activation barrier
U . Ec defines the electric field level of the critical current density jc . The timescale for
reaching a nearly stationary state t0 depends strongly on the sample size, sample geometry and
sweep rates of Hex and can cover a range of t0 of 10−5 up to several 10 (ten) seconds (see e.g.
Jooss et al (2001b)).
For transport currents, a similar time constant for the development of a quasi-stationary
state:
t0 ≈ µ0 W 2 j1 /E,
(14)
was defined by Gurevich and Friesen (2000), with j1 = jc s, where s = d ln j/d ln t represents
the dimensionless flux creep rate. Taking a standard electric field criterion of Ec = 1 µV cm−1
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Ch Jooss et al
for jc , a characteristic extent of an area with uniform current of L = 1 mm, jc = 1011 A m−2
and s = 0.05, one obtains t0 ≈ 60 s at low temperatures. This timescale can become much
smaller, e.g. for a sample with a high density of large defects, but also can reach very long times
up to hours, e.g. in the vicinity of a planar defect (Gurevich and Friesen 2000). Experimental
results of several 100 ms for relaxation times of pulsed transport currents in thin-film stripes
are obtained by Bobyl et al (2001).
When comparing critical currents which are measured by different measurement methods,
the different timescales or electric field levels have to be taken into account. The jc s which
are determined by MOI in magnetization experiments on typical timescales of several seconds
correspond to electric field criteria which are several orders of magnitude smaller compared
to those of resistively measured transport currents.
In the following, we focus on the description of the relaxation process of averaged
properties on the mesoscopic level which are observed by typical magneto-optical experiments
and start from Maxwell’s equation
Ḃ (r ) = −∇ × E (r ).
(15)
The electric field E during the relaxation process is created by the drift velocity field v (r ) of
the vortices. It is given by
E (r ) = v (r ) × B (r ),
(16)
and the combination of the two equations leads to the equation of motion for the magnetic flux
density
Ḃ (r ) = −∇ × (v (r ) × B (r )).
(17)
The drift velocity for thermally activated hopping of the vortices follows an Arrhenius law:
v (r ) = r0 ω0 e−U (j (r))/kB T ,
(18)
where r0 represents the averaged hopping distance and ω0 is a microscopic attempt frequency.
Generally, the activation barrier U (j ) for the flux movement depends on many different
factors such as the microscopic distribution of the pinning potential u, the anisotropy of the
superconductor and also on the local flux and current density. Different models for U were
given by various authors (Anderson 1962, Anderson and Kim 1964, Beasley et al 1969, Zeldov
et al 1989). The development of the vortex glass (Fisher 1989) and collective creep (Feigel’man
et al 1989, 1991) theories for the HTS result in a power-law behaviour for U (j ). In addition
to the models which assume a single typical value of the activation barrier U0 , a spectral
distribution of the activation energy was suggested by Hagen and Griessen (1989) and Theuss
and Kronmüller (1994). Direct evidence for a spatial variation of the activation energy was
found by Warthmann et al (2000); for reviews see Blatter et al (1994) and Yeshurun et al
(1996).
3. Flux imaging using magneto-optics
The magneto-optical Faraday effect represents an excellent method for the space-resolved
measurement of the magnetic flux density distribution of a superconductor. Since up to now
a significant Faraday or Kerr effect of superconductors has not been observed, one has to
use MOLs which are placed or evaporated on top of a superconductor. For flux imaging in
superconductors, all the different magneto-optical materials used are based on the Faraday
effect. The preparation and the properties of paramagnetic Eu chalcogenides and halogenides
are described and compared extensively in the literature (Hübener 1979, Schuster et al 1991,
Koblischka et al 1995). We focus here on two different MOLs: paramagnetic EuSe and
Current distributions in high-Tc superconductors
669
Figure 6. The basic principle of the measurement of the magnetic flux distribution of
superconductors by the magneto-optical technique.
ferrimagnet-doped iron garnet films with in-plane magnetization which are the most common
and most important for the magnetic flux visualization in superconductors; for an overview
see also Polyanskii et al (1999, 2001c). A short introduction is given into their application in
making quantitative measurements of the magnetic flux and supercurrent distributions.
3.1. Faraday effect
In materials with longitudinal optical birefringence, the polarization vector of an incident
linearly polarized light beam propagating over a length l parallel to a magnetic field H is
rotated through an angle α(H ). The two eigenmodes of light propagation in the crystal are
right and left circularly polarized light. The difference in real index of refraction between
these modes n = nL (ω, H ) − nR (ω, H ) gives rise to the Faraday rotation
ωl
n.
(19)
2
The difference in real index of refraction n is directly proportional to the expectation value
of the magnetic moments µz along the propagation axis (z-axis) (Suits et al 1966). For
paramagnetic materials or for the virgin curve of materials with spontaneous magnetization,
one has Mz = χ Hz for small magnetic fields. In this case one may write equation (19) as a
Taylor series in Hz and the linear approximation gives
α=
α = V (ω)lHz .
(20)
The Faraday rotation depends on the traversed length l of the MOL, the magnetic field
component Hz parallel to the light beam and a material-specific and frequency-dependent
constant V (ω), the so-called Verdet constant. This effect was first observed by Faraday (1846).
Since the Faraday rotation α is directly proportional to the magnetization component Mz
parallel to the beam, a magnetic domain structure of a magnetically ordered material may
strongly disturb the application of a MOL as a field-sensing element. EuSe is a magnetic
semiconductor which exhibits ferromagnetic ordering for T < 2.8 K, antiferromagnetic
ordering for 2.8 K < T < 4.6 K and paramagnetic behaviour for T > 4.6 K (Hübener
1979). The huge Faraday rotation of up to 110◦ µm−1 (T = 4.5 K, µ0 Hex = 1.15 T and
λ = 560 nm) and saturation fields larger than 1.2 T (Schoenes 1975) makes this material well
suited for magneto-optical imaging in a large range of magnetic fields. From the viewpoint
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Ch Jooss et al
Table 1. Magneto-optical characteristics of EuSe and iron garnets with the in-plane magnetization
axis.
Magnetic resolution
Magnetic saturation
Spatial resolutiona
Temperature range
Verdet constant V (ω)d
EuSe
(Bi, Lu, Ga, Y)–iron garnets with
in-plane easy magnetization axis
1 mT
1.1 T
500 nm
15–20 K
0.096◦ mT−1 µm−1
10 µT
100–300 mT
5 µm (370 nm)b
400–600 Kc
0.008◦ –0.04◦ mT−1 µm−1
a
The optimal spatial resolution which is limited by the MOL itself is given here.
A submicron resolution was achieved by pressing the iron garnet onto a hard disk drive of a
computer with perpendicular domains (Grechishkin et al 1996).
c The ferrimagnetic compensation temperature depends on the doping of the iron garnets; typical
values are given.
d EuSe: λ = 546 nm (maximum V (ω). Iron garnets: V (ω) depends on the dopants and on H . A
k
typical range of values is given for λ ≈ 450–550 nm for fields far below the saturation field.
b
of magnetic ordering, the application of EuSe is suitable for flux imaging for T > 4.6 K, in
order to avoid magnetic domain structures. Practically, it is used for T 4.2 K. Due to the
strong decrease of its Verdet constant with increasing temperature, the application is limited
to temperatures <15–20 K.
In contrast to the case for the Eu chalcogenides, the change of the magneto-optical
characteristics of iron garnet films with temperature is very small in the temperature range
of HTS but magnetic ordering gives rise to complications due to magnetic domains which may
disturb the magnetic field mapping and magnetic hysteresis. Iron garnets are compounds with
the general formula {Me3+ }3 [Fe3+ ]2 [Fe3+ ]3 O2−
12 , where Me is a trivalent metallic ion. Using
Lu3+ one gets ferrimagnetic order with uniaxial anisotropy. In-plane magnetization is obtained
by doping with Bi, Ga, Lu and Y under special conditions (Grechishkin et al 1996). Bi doping
increases the Faraday rotation and one obtains specific Faraday rotations in the range of 7◦ –
9◦ µm−1 (at room temperature and λ = 633 nm) (Okuda et al 1990, Adachi et al 2000).
For measurements of the Faraday rotation spectra of Bi-doped iron garnets, see (Helseth et al
2001). Iron garnet films which are used for flux imaging must have a small defect density and
are therefore grown by liquid phase epitaxy on single-crystalline Gd–Ga garnet substrates.
Films with perpendicular anisotropy exhibit a bubble or labyrinth-like domain structure
with domain sizes in the range of 4–50 µm (Polyanskii et al 1989, Indenbom et al 1990). At
λ = 426 nm specific Faraday rotations parallel to the magnetization vector up to 4◦ µm−1 in
Y3−x Bix Fe5 O12 are obtained at room temperature (Simsa et al 1984), which depend strongly
on the Bi content. If the polarized light beam is propagating parallel or antiparallel to M ,
the Faraday rotation has different signs in different domains. Consequently, the linear relation
between magnetic field and α according to equation (20) is not applicable. However, the
observation of the domain distribution in the iron garnet film allows an indirect visualization
of the locally applied magnetic field (Polyanskii et al 1990a, 1990b, 1990c, Szymczak et al
1990, Indenbom et al 1990, Gotoh et al 1990, Belyaeva et al 1991b, Gotoh and Koshizuka
1991).
These characteristics can be substantially improved by the application of films with inplane easy magnetization direction (Dorosinskii et al 1991); see also Belyaeva et al (1991a),
Vlasko-Vlasov et al (1991), Dorosinskii et al (1992). If a magnetic field Hz is applied normal
to the film plane, the in-plane magnetization vector M is rotated out of the film plane by an
angle
Current distributions in high-Tc superconductors
671
Hz
.
(21)
Hk
Hk represents the anisotropy field of the film. Similarly to the paramagnetic MOLs, a linearly
polarized light beam propagating normal to the film plane is rotated by an angle α which is
proportional to Mz . The presence of a spontaneous magnetization vector, however, causes a
non-linear relation between the normal component Mz and the external magnetic field Hex
which gives a Faraday rotation of
Hz
.
(22)
α = cMz = cMs sin φ = cMs sin arctan
Hk
Ms indicates the spontaneous magnetization of the ferrimagnetic film and c is a materialspecific constant similar to Verdet’s constant. Note that equation (22) is only applicable if
magnetic hysteresis is completely negligible. In order to obtain a high magnetic resolution and
a reasonable field range without saturation, the anisotropy field Hk of the iron garnet should
be larger than the measured field strengths. Typical values are µ0 Hk ≈ 100–300 mT; the
composition may be adjusted to extend µ0 Hk up to 1 T (Grechishkin et al 1996).
A serious problem for the calibration is the presence of magnetic stray fields parallel to
the superconductor surface (Johansen et al 1996). Due to the coupling of these in-plane field
components Bx or By to the magnetization vector Ms , the rotation angle φ can be suppressed
significantly. In this case, Hk in equation (21) must be replaced by Hk + Hx . A calibration
procedure which takes this in-plane coupling into account is successfully applied by Johansen
et al (1996). Since the in-plane components of the self-field increase with the lateral sample
size W , the calibration errors due to the in-plane coupling of Ms can be minimized by reducing
W and using indicators with relatively large Hk , such that Hx Hk .
Compared to the case for the uniaxial garnets, much less has been reported on the specific
Faraday rotation of the garnets with the in-plane easy magnetization axis. Grechishkin et al
(1996) give a value of 1.2◦ µm−1 at room temperature without noting the light wavelength.
Goa et al (2001) report a Verdet constant of V = 0.0083◦ µm−1 mT−1 at λ = 546 nm and small
magnetic fields. Our own measurements at 4.2 K reveal specific Faraday rotations between 3
and 10◦ µm−1 in the saturated state (φ = π/2) and V up to 0.04◦ µm−1 mT−1 at small fields
(λ = 500 nm).
φ = arctan
3.2. Experimental set-up and resolution
Magneto-optical experiments require a polarized light microscope, an optical cryostat, an
electromagnetic coil for the generation of magnetic fields and some kind of image recording
system such as a camera, a video system or for digital image processing a CCD camera
(charge-coupled device). A possible basic experimental arrangement is depicted in figure 7.
The polarized light microscope consists of a stabilized light source with sufficient power, a
polarizer, an analyser and optical components to project the image plane of the MOL into
the image recording system with various magnifications. These components, especially the
objective lenses, should have small Verdet constants to avoid disturbing Faraday rotations due
to magnetic stray fields of the coils. In addition, they should not depolarize the polarized
light beam. Different experimental set-ups which were used for the investigation of HTS are
described by several authors (Moser et al 1989, Indenbom et al 1990, Gotoh et al 1990, Forkl
et al 1990, Belyaeva et al 1991b, Schuster et al 1991, Forkl 1993, Vlasko-Vlasov et al 1993b)
and for large-area magneto-optics by Kuhn et al (1999b); for details of an optical cryostat, see
e.g. Greubel et al (1990).
Generally, magneto-optical measurements are performed in reflection mode. Due to the
strong absorption of the HTS in the frequency range of visible light, it is necessary to use
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Ch Jooss et al
Figure 7. A sketch of the basic experimental arrangement for magneto-optical imaging of magnetic
flux patterns in superconductors. The polarized light microscope has the following components:
(1) stabilized light source, (2) collimator, (3) edge filter for infrared suppression, (4) polarizer,
(5) semi-reflecting mirror, (6) objective, (7) covering glass of the cryostat (Suprasil), (8) sample
covered with a MOL and (9) analyser.
a)
c)
b)
10 µ m
Figure 8. Magneto-optical images of vortices in a NbSe2 single crystal at 4.0 K (a) after cooling
in the Earth’s field (≈0.1 mT) and (b) after application of µ0 Hex = 0.3 mT. In (c), the vortex
dynamics after a small increase of the external field in a time window of 1 s is observed. Scale bar:
10 µm. (Goa et al 2001)
Current distributions in high-Tc superconductors
673
mirror layers in order to obtain a better reflectivity. Since polycrystalline EuSe may be easily
evaporated directly on the top of the superconducting sample—whereas single-crystalline iron
garnets have to be grown epitaxially on suitable substrates, two different alternatives for the
mirror layers have to be used: in the case of EuSe, a 200 nm thick layer of aluminium or gold
may be deposited directly on the sample surface—followed by the evaporation of the MOL.
Although the properties of the superconductor may be influenced by a deposited metallic
layer, one has the advantage of a higher spatial resolution limited only by properties of the
polarization microscope. In contrast, for iron garnets, the mirror layer is directly deposited
onto the MOL. In this case, the spatial resolution of the measurement is also limited by the
distance between the MOL and the superconductor surface. For a magneto-optical technique
with a EuS film on a glass plate, see Bujok et al (1993) and Brüll et al (1991).
For the determination of the optimum film thickness of the MOL, the light which is
reflected at the upper surface of the MOL without passing through the magneto-optical film
has to be taken into account. For EuSe, the rotation angles are typically smaller than 1◦ and
thus the non-rotated component of the light beam is strongly disturbing the measurement.
This light beam can be cancelled out by destructive interference (Hübener et al 1979). For a
wavelength of 546 nm of the polarized light, where EuSe has the largest V (ω), the phase and
the amplitude condition for destructive interference gives a optimum film thickness for EuSe
of 248 nm (Schuster et al 1991).
In case of iron garnet MOLs, the specific Faraday rotation is significantly larger and
up to now a cancelling of the non-rotated beam has not been performed. A typical fieldsensing element consists of an ≈500 µm thick Gd3 Ga5 O12 substrate, a ≈ 2–8 µm thick doped
iron garnet film and reflecting titanium or aluminium layers with thicknesses of ≈100 nm.
The spatial resolution of this system is limited by the film thickness of the iron garnet, the
measurement height, when the MOL is placed onto the surface of the superconductor and the
resolution of the optical microscope. Typically, a spatial resolution of ≈5 µm is achieved,
depending on the individual iron garnet properties and the value and length scale of the field
gradients which are creating the magneto-optical contrast.
Recently, the magneto-optical imaging of individual vortices in NbSe2 was achieved by
the Oslo group (Goa et al 2001) (see figure 8). This has become possible because of significant
improvements of the magneto-optical imaging system, such as mounting the objective directly
on the cryostat, using Glan–Taylor polarizer/analysers and a Smith beam splitter. This opens
up a new field of real-time single-vortex imaging by magneto-optical measurements.
3.3. Calibration of the flux density
Magneto-optics has been used over many years for a qualitative visualization of the magnetic
flux distribution in superconductors. The progress in digital image recording and image
processing by computers opened the possibility of a detailed quantitative analysis of the
magnetic flux density distribution at the surface of a superconductor (Jooss et al 1996a,
Johansen et al 1996). The calibration of the measured light intensity distribution I (x, y)
into a magnetic flux density distribution Bz (x, y) requires a quantitative description of the
polarization effects of the MOLs as well as the transfer of a polarized light beam through a
polarization microscope. Disturbing additional effects on the polarization vector in a nonideal optical system, such as inhomogeneous illumination of the MOL or polarization effects
of optical lenses and mirrors, have to be corrected.
The light intensity I , reflected out of a MOL with thickness d and absorption coefficient
γ , is I = I0 e−2γ d , where I0 represents the intensity of the incident polarized light beam. For
an ideal polarization microscope, the light intensity I of a light beam traversing the polarizer,
Ch Jooss et al
B
674
0
Bk
rad
s
00
∆α
0
rad
Intensity I(a,b)
Figure 9. A plot of the measured B = µ0 Hex versus I curve for a iron garnet film with planar
anisotropy. For the fit, equation (25) was used. The fit parameters obtained are shown in the plot.
MOL and analyser is given by the Malus law:
I = I0 e−2γ d sin2 (α + α) = I sin2 (α + α),
(23)
where α denotes the deviation of the polarizer and analyser from the crossed orientation (90◦ ).
The Faraday rotation α is determined by equations (20) and (22) respectively, depending on the
MOL used. In principle, equation (23) allows a calibration of the magneto-optically measured
light intensity distribution I (x, y) to a magnetic flux density distribution Bz (x, y) at a plane
z = constant above the superconductor. However, for the application of equation (23) the
deviations due to the non-ideal optical set-up have to be taken into account.
Due to the finite degree of polarization of the polarization filters and depolarization effects
of the optical components, there remains a small residual light intensity I1 even for α +α = 0.
If the Faraday rotation α is small, this background intensity I1 has to be subtracted from the
measured intensity I . Further, the inhomogeneous illumination of the MOL due to non-perfect
(e.g. Gaussian) spatial distribution of the polarized light intensity has to be taken into account.
Indeed, I (x, y) and I (x, y) are functions of the coordinates x, y within a plane perpendicular
to the beam. Thirdly, the Verdet constants of the optical components of the microscope do
not vanish. Magnetic stray fields of the coils may lead to an additional Faraday rotation and
contributes to α. A possibility for diminishing stray-field contributions of the coil is to
use soft magnetic screenings around the optical system. Further, it is necessary to use special
glasses with low Verdet constants for covering the optical cryostat and, in addition, tension-free
lenses in the objective to minimize depolarization effects. Fourthly, the camera characteristics
have to be taken into account. In the following, we assume a ideal linear response of the
camera which is realized to a high extent by a CCD, if applied between the noise level and the
blooming limit.
Taking all above-described effects into account, we get two different functions for the
calibration of EuSe:
I (x, y) − I1 (x, y)
1
Bz (x, y) =
arcsin
+ α ,
(24)
Vd
I (x, y)
and for the iron garnet MOLs with in-plane anisotropy:
Current distributions in high-Tc superconductors
675
Table 2. Summary of typical calibration problems and errors for the quantitative determination of
Bz (x, y) in magneto-optical measurements and strategies for their solution. Typical examples are
shown in some of the figures in this article as referenced in the second column.
Calibration problem
Effect
Solution
Inhomogeneous
illumination I (x, y)
Distortion of ∇Bz (x, y)
Background subtraction.
Spatially dependent calibration.
Hex -dependent
polarization effects
in optical components
Distortion of ∇Bz (x, y);
see figure 21
Shielding of Hex
from the optics.
Full analysis of I (x, y, Hex ).
In-plane coupling
of H to Ms
Distortion of φ, α
and Bz (x, y);
see figure 14(a)–(d)
Jumps in Bz (x, y);
Correction of the in-plane fields.
Ensure Hx Hk .
Magnetic domain
structure of MOL
Magnetic saturation
see figure 14(e)
Distortion of ∇Bz (x, y)
Sign change of Bz (x, y)
Imaging of |Bz (x, y)|
Defects in the MOL
Spots, lines;
see figure 20
Paramagnetic MOLs.
Perpendicular illumination.
Materials optimization.
Adjustment of Hk
to the experiment.
Differential imaging
with ±α.
Improvement of
quality of MOLs.
I (x, y) − I1 (x, y)
1
Bz (x, y) = Bk tan arcsin
+ α . (25)
arcsin
cMs
I (x, y)
Practically, the determination of calibration curves at each images position (x, y) is impossible.
One determines a calibration curve at one position (x = a, y = b) within the image by
measuring I (Hex ) for an external field Hex z. Afterwards, the functions (24) or (25) are
fitted to the measured I (Hex ) curves with the fit parameters I and α. I1 (x, y) is a background
image, which has to be subtracted from the I (x, y) data and the spatial distribution of the light
reflected from the MOL I (x, y) may be determined by normalization of the background
image I1 (x, y) to I (x, y) = I1 (x, y)I (a, b)/I1 (a, b). Here (x = a, y = b) indicates the
position within the image, where the calibration curve has been recorded. Having determined
all parameters and the distributions I1 (x, y) and I (x, y), a spatially dependent calibration
according to equations (24) or (25) is applied to the measured MO contrast I (x, y), in order
to obtain the flux density distribution Bz (x, y).
Table 2 gives an overview on typical calibration problems and errors for the determination
of Bz (x, y) from magneto-optically measured light intensity distributions.
3.4. Determination of supercurrents
Imaging of the magnetic flux density distribution at the surface of superconductors provides
qualitative insights into flux pinning and supercurrent distributions. In many cases, however,
it is desirable to obtain quantitatively the magnitude and the directions of the flowing shielding
and critical current densities. The general relation between the measured flux density
distribution and the flowing supercurrent density j is given by Ampère’s law
µ0 j = ∇ × B .
(26)
In real flux imaging experiments, one has flat samples with finite sizes and the flowing
supercurrent generates magnetic self-fields with strongly curved field lines. Consequently,
676
Ch Jooss et al
the application of the original Bean model which is valid for long samples in parallel fields is
not possible (Frankel 1979, Theuss et al 1992). It relates the current density to gradients in
the parallel component of B and completely neglects the gradients of the other (curved) field
components according to equation (26). However, for the application of Ampère’s law one has
to measure the spatial distribution of all three components of B = (Bx , By , Bz ). Even if it is
in principle possible to measure in-plane components of B by applying the transverse Faraday
effect (see e.g. Dorosinskii and Polyanskii (1993b)), it is in practice difficult to measure their
gradients by means of magneto-optics.
As we will show in the following, it is sufficient for most practical situations to measure
the normal component Bz at the sample surface if one proceeds from the local relation (26) to
the integral relation between j and B . The Biot–Savart law, representing the reversed Ampère
law, gives the integral relation between the current density and the magnetic field. For the
measured Bz -component of the magnetic flux density one has
jx (r )(y − y ) − jy (r )(x − x ) 3 µ0
d r.
(27)
Bz (r ) = µ0 Hex +
4π V
|r − r |3
The measurement of the perpendicular flux density distribution Bz (x, y, z) above the
superconductor surface enables the determination of the planar current density j (x, y, z) =
ex jx (x, y, z) + ey jy (x, y, z). Equation (27) may be used in two different ways for the
quantitative analysis of the current distribution: either one uses models for the current
distribution j (x, y) and compares the calculated Bz (x, y) distributions with the measured
ones; or one directly inverts equation (27) by numerical methods. The latter provides a modelindependent method for the determination of j (x, y). In order to obtain an unambiguous
mapping of the measured Bz (x, y) to j (x, y), the following additional restrictions have to be
considered:
(a) The determination of two current density components jx and jy from one measured field
component Bz requires an additional relation between jx and jy , notably,
∇ · j = 0.
(28)
For an isolated superconductor where the supercurrents are created in a magnetization
experiment in external magnetic fields, this is directly evident: the current streamlines are
closed and one has no sources and drains for the current. Equation (28) is, however, also
valid for transport currents which are transmitted through a superconductor. Due to the
limited measurement area for Bz (x, y) in the x–y plane equation (28) cannot be applied
at the image borders of the magneto-optical images. We come back to this topic below.
(b) In principle, it is possible to determine the z-dependence of the planar supercurrent by the
measurement of Bz at different distances from the superconductor surface. By means of
magneto-optics the Bz (x, y, z) distribution is usually measured in one plane z = constant
above the sample and the z-dependence of j (x, y) is therefore not accessible. In HTS
thin films with thicknesses d < λ, however, the z-dependence of j (x, y) is negligibly
small.For large magnetic fields the current density may vary over the intervortex distance
√
a = 20 /( 3B) and the criterion d < λ should be replaced by d < a. For thicker
samples one may introduce a thickness-averaged current density:
1
J=
(29)
dz j (x, y, z),
d
which will be called in the following the sheet current density. If the sample thickness d
significantly exceeds the measurement height h, there is mainly sensitivity to the surface
currents within a surface layer of thickness τ . The ratio Bz,τ /Bz,d of the stray field being
Bz,d / B z,τ
Current distributions in high-Tc superconductors
677
h = 1 µm
h = 5 µm
h = 10 µ m
Figure 10. The ratio Bz,τ /Bz,d of the stray field being created by a current string within a surface
layer τ to the total stray field of a current string flowing in the full sample thickness d > τ > h.
With decreasing h the measurement is becoming more sensitive to the surface currents.
created by a current string within a surface layer τ to the total stray field of a current string
flowing in the full sample thickness d > τ > h is given by
Bz,τ
2
τ
.
(30)
= arctan
Bz,d
π
2(h(h + τ ))1/2
Here, it is already taken into account that the position of the maximum of Bz is shifted in
the x–y plane when increasing τ . Figure 10 shows that ≈70% of the stray field is created
by surface currents within τ = 20 µm in thick samples d h, if a typical h = 5 µm is
used.
(c) The perpendicular component of the supercurrent density jz is not accessible at all by
measuring the normal flux density component Bz . In thin films (d < λ) the jz -component
is always small, even if the applied external field is tilted to the normal axis of the film. For
thick samples one has to carry out magneto-optical imaging on different sample sides in
order to determine all current density components. Note that due to the strong anisotropy
of the layered HTS, the ratio of critical current densities jc,c (parallel to c) and jc,ab (in
the a–b plane) is of the order of 10−1 for YBCO and 10−3 –10−4 for Bi-2223 (Senoussi
1992).
As already mentioned above, there are two basic methods for extracting the planar
supercurrent density by means of equation (27). The first one is to use a model for the
current distribution and compare the calculated flux distribution with measured flux densities
(e.g. Frankel 1979, Theuss et al 1992, Knorrp et al 1994, Schuster et al 1994a, Forkl and
Kronmüller 1995, Polyanskii et al 1996, Jooss et al 1996a). The models used for the current
distribution are partly based on the Bean model (Bean 1964) extensions to flat geometries.
Other models use Maxwell’s equations together with a current–voltage curve of a non-linear
conductor to calculate the flux and current distributions; see e.g. Brandt (1995c), Schuster et al
(1995a, 1996a). These models will be described and compared to flux imaging experiments
in the later sections of this paper.
We will come back now to the second basic method for determining j (x, y) from the
measured Bz (x, y) by inversion of equation (27) (Grant et al 1994, Niculescu et al 1996,
Brandt 1995b, Wijngaarden et al 1996, Johansen et al 1996, Pashitskii et al 1997, Jooss et al
1998a). Despite the restrictions discussed before, this second basic method represents a modelindependent determination of the current density distribution. It is beyond the scope of this
678
Ch Jooss et al
paper to compare all the different methods for the inversion of the Biot–Savart law. The reader
will find some aspects in Wijngaarden et al (1997) and Jooss et al (1998a). Here, a method
extending the method of Roth et al (1989) is summarized which allows simultaneously high
accuracy, high spatial resolution and low calculational effort.
On the basis of equation (28), one may derive the two in-plane current density components
from one scalar function g(x, y) by jx (x, y) = ∂g(x, y)/∂y and jy (x, y) = −∂g(x, y)/∂x
(Brandt 1992a), where g has the meaning of a local magnetic moment or a density of tiny
current loops. The perpendicular magnetic flux density is related to g by a non-local relation
which is obtained from the Biot–Savart law, equation (27):
Bz (r ) − µ0 Hex = µ0 Kg (r − r )g(r ) d3 r ,
(31)
where the integral kernel Kg possesses translational invariance in the x–y plane. It is given by
Kg =
1 2z2 − (x − x )2 − (y − y )2
.
4π
|r − r |5
Applying two-dimensional Fourier transformations
∞
∞
B̃z (kx , ky ) =
dx
dy Bz (x, y)ei(kx x+ky y)
−∞
−∞
∞
∞
g̃(kx , ky ) =
dx
dy g(x, y)ei(kx x+ky y) ,
−∞
(32)
(33)
−∞
and extending the integration in equation (31) to an infinite plane, one obtains by the application
of the convolution theorem
B̃z (kx , ky , h) = µ0 K̃g (kx , ky , h, d)g̃(kx , ky ).
(34)
The integral over the g-distribution times an integral kernel in equation (31) reduces to a simple
product of two functions in k-space. After performing the z -integration and assuming that the
current density does not depend on z , the Fourier-transformed integral kernel is given by
kd
−kh
K̃g = e
sinh
.
(35)
2
Here, h is the measurement height above the superconductor and d the thickness of the
specimen. The inverse relation between g̃ and B̃z is simply given by
g̃ =
B̃z
µ0 K̃g
.
(36)
The application of Fourier transformation and the convolution theorem leads to a periodic
continuation of the field and current distribution in the infinite x–y plane. In order to avoid
a distortion of the j (x, y) determined, by the periodic continuation of the magnetic field, the
magneto-optical measurements should cover an area larger than the superconductor. If the
decay of the stray field of the superconducting sample is measured up to a distance of half of
the lateral sample extension, the periodic supercells are completely decoupled and the error
in the current density determined is less than 0.1%. However, in some cases, such as a stripe
with a transport current, a periodic continuation is strongly desirable, since the violation of
equation (28) at the image borders can be avoided. For further details of this inversion scheme,
see Jooss et al (1998a).
Current distributions in high-Tc superconductors
679
4. Current distribution and sample geometry
If a magnetic field at the surface of a type-II superconductor exceeds Hc1 in magnetization or
transport experiments, magnetic flux lines enter the superconductor from all surfaces. Pinning
at material defects leads to the generation of magnetic flux gradients towards all superconductor
surfaces. Whereas in fact the development of the critical state during flux penetration is a full
three-dimensional problem, only the flux penetration from a surface which is oriented parallel
to the applied external field Hex is considered in the original Bean model. This can be justified,
e.g. for long cylinders in parallel fields, where the surface areas oriented normal to Hex are
small and, therefore, the curvature of the field lines is only important in the small surface
regions at the ends of the cylindrical sample.
However, the sample geometry assumed in the Bean model is not realized in typical
HTS magnetization or transport experiments. Measurements are usually performed on singlecrystalline platelets, tapes, thin films or melt-textured superconductors, where the sample
thickness d parallel to the z-axis is comparable to or much smaller than the sample width W in
the x–y plane. In this case, the curvature of the field lines which leads to a three-dimensional
‘lens-shaped’ flux penetration has to be taken into account. A simpler case, however, is
represented by thin films (d λ) in a perpendicular external field Hex z, such that the
dependence of the critical current along z can be completely neglected and analytical solutions
of the flux and current distribution are already obtained in the simplest case of one-dimensional
currents in thin strips and discs.
HTS ceramics often exhibit complex shapes due to cracks, arbitrary surfaces, corners or
boundaries to normal phases. Due to the boundary condition that the normal component of
the supercurrent has to vanish at all surfaces and the presence of large, geometry-dependent
magnetic stray fields, the sample shape is strongly influencing the current distribution. In the
critical state, where jc = constant over the whole sample, the vector field j with constant
modulus splits up into domains with uniform parallel current flow. These domains with
different j -orientations and different directions of the magnetic flux gradients are separated by
discontinuity lines. Much more complicated is the current distribution in the partly penetrated
state, where j = constant is not fulfilled. Here, the magnitude as well as the orientation of j
is changing during flux penetration. In addition, the electric field pattern during flux flow and
flux creep affects the current distribution, and the magnitude as well as the distribution of the
supercurrents has to be considered as a function of the time window or electric field level of
the experiments.
This section is organized as follows: first we consider the simplest case of one-dimensional
current density distributions of an infinitely extended thin strip with d < λ, where analytical
solutions for the flux and current distributions are available; section 4.1. Then, we continue in
section 4.2 with the consideration of rectangular and arbitrarily shaped thin films in the critical
state, where the current distribution can be easily constructed. Further, the process of flux
penetration and partly penetrated states are considered for rectangular and arbitrarily shaped
thin films in section 4.3. Some aspects of the modification of j (x, y) due to inhomogeneous
electric fields (induced by flux motion) are considered in section 4.4. Section 4.5 goes into
the details of current distributions which continuously change their orientation (in contrast
to discontinuity lines). They are important due to their high electric field level during flux
movements. The final part (section 4.6) describes results for samples with finite thickness,
where the z-dependence of the current density cannot be neglected.
We focus here on the geometry dependence of the flux and current distributions and
neglect all further influences such as reversible properties, grain boundaries, local magnetic
field dependence of jc , inhomogeneity and anisotropy which are considered in sections 5–7.
680
Ch Jooss et al
Figure 11. A sketch of the geometry of the YBCO strips and choice of coordinates.
4.1. Thin strips
For thin superconductors in a perpendicular external magnetic field, the superposition of the
stray field of the supercurrents with Bex strongly changes the field distribution assumed in the
original Bean model and causes a curvature of the field lines of B . We consider here a thin
strip of thickness d parallel to z and width W d extended infinitely in the x-direction. For
this geometry we obtain a one-dimensional current density jx (y) flowing along the strip edges.
The Lorentz force density is fL = jx (Bz ey − By ez ). The z-component of the driving force
is due to the curvature of the magnetic field lines. Thus, the flux not only penetrates in the
direction perpendicular to the applied field as in the parallel limit described in section 2. The
flux penetration takes place in two dimensions, from the strip edges as well as from the surface
of the strip. Note that in thin films d < λ, the shapes of the vortices in general differ from
the shape of the curved field lines because of the boundary condition jv,z = 0 at the surface
(Brandt and Indenbom 1994).
For W d, the curvature dominates the perpendicular flux gradients and the critical
current density derived from Ampère’s law is approximately given by the gradient of the inplane component jc ≈ ∂By /∂z (Theuss et al 1992) which, however, cannot be measured by
means of magneto-optics. In order to determine critical current densities in the perpendicular
limit from the measured normal component Bz (x, y) of the flux distribution, one has to use
the Biot–Savart law instead of Ampère’s law (see also section 3.4).
For a thin long strip with jc = constant and thickness d → 0, the problem of flux
penetration was solved analytically by Brandt et al (1993). The similar problem of the flux
and current distribution of a transport current had already been solved by Norris (1970). If
Hex z is increased from the ZFC state, one obtains for the current density in a strip
 y

Q |y| W/2

 jc |y|
(37)
jx (y) =
2jc
yc


|y|
<
Q,
arctan

π
(Q2 − y 2 )1/2

|y|c
jc µ0


arctanh
|y| > W/2



(y 2 − Q2 )1/2
 π
2
Bz (y) = jc µ0
(38)
(y − Q2 )1/2

Q |y| W/2
arctanh


|y|c
π



0
|y| < Q,
with c = 2((W/2)2 − Q2 )1/2 /W and Q indicates the penetration depth of the flux front
measured from the strip centre. The penetration depth of the magnetic flux measured from the
Current distributions in high-Tc superconductors
681
strip edge is P = W/2 − Q with
1
W
.
Q=
2 cosh[πHex /jc d]
The penetration field for a strip in the perpendicular limit is (Brandt 1996a)
2W
jc d
Hp =
ln
,
π
d
(39)
(40)
which may be used to estimate jc from MO measurements but the precision of this method is
not very high.
Equation (38) gives the correct flux distribution only in the limit d = 0 directly at the
surface of the superconducting film and diverges at the strip edges. For comparison with
experiments one may use the current distribution of equation (37) for calculating the flux
distribution Bz (x, y, h) at a distance h above the surface of a superconductor with finite
thickness d λ by numerically integrating the Biot–Savart law
µ0 W/2 d/2 jx (y , z )(y − y )
dy dz ,
(41)
Bz (x, y, h) = µ0 Hex +
2π −W/2 −d/2 (y − y )2 + (h − z )2
or one may use
µ0 W/2 Jx (y )2y Bz (x, y) = µ0 Hex +
dy ,
(42)
2π −W/2 y 2 − y 2
which leads to equation (38). Equation (42) represents the limit limh,d→0 of equation (41),
where the current density is replaced by the sheet current according to equation (29).
As examples, figures 13(a) and (b) depict results from equations (37) and (41) together
with experimental data from figure 12. In comparison to the longitudinal Bean model, there
are characteristic differences like the (Meissner) screening current js < jc in the flux-free
region (see section 5.5), the large slope of Bz (y) at |y| = Q and the peak in the flux density at
the strip edges. For a thin disc, a similar analytical model for the radial flux penetration was
given by Mikheenko and Kuzlovlev (1993).
Up to now, all considerations apply to the Bean model, i.e., to steplike current–voltage
characteristics, E(j < jc ) = 0 and E(j > jc ) = ∞. For smooth E(j ), occurring at finite T
in HTS, the resulting j , H , M and B depend on the time t, i.e. the ramp rate Ḣex (t) = dHex /dt
enters or, if Ḣex = 0, the flux distribution varies with time due to flux creep. The equation
of motion for the sheet current J (y, t) (strip) or J (r, t) (disc) is then derived from Faraday’s
induction law as follows. (For the dynamics, we assume that j = J /d = constant over the
sample thickness and consider the limit d → 0.)
r
y
When the magnetic flux φ(y) = µ0 L 0 H (y ) dy or φ(r) = 2π µ0 0 H (r )r dr threading part of the strip (of length L) or disc varies with time, it induces an electric field
E(y, t) = φ̇(y, t)/L or E(r, t) = φ̇(r, t)/2π r along the boundary of this area. Inserting here
equation (42) and E = ρ(j )j one gets the integro-differential equations for the sheet currents
J (y, t) (strip) and J (r, t) (disc) (Brandt 1993, 1994a, 1994b):
1
(43)
J˙(y , t) K(y, y ) dy .
J (y, t) = τ 2πy Ḣex (t) +
0
Here τ = τ (J ) = 0.5W dµ0 /2πρ(J ) is a relaxation time, which in general may depend on
B and J (via ρ) and on the position r (via ρ, B or J or if the thickness d depends on r ). In
equation (43) the half-width W/2 is chosen as unit length and the integral kernel is
y − y .
(44)
K(y, y ) = ln y + y 682
Ch Jooss et al
Figure 12. Magneto-optically (EuSe) detected flux distributions of a strip of width W = 200 µm
and thickness d = 250 nm at T = 5 K and µ0 Hex = 20, 43, 64 and 92 mT. Due to the distortion
of the evaporated MOL the strip edges appear as horizontal dark lines.
The equation (43) may be integrated over time t on a computer by a method described in
Brandt (1994b). From the resulting sheet current the magnetic flux density Bz (y, t) is obtained
from equations (41) or (42).
The use of the model current–voltage law
E(j ) = [j/jc (B, r )]n Ec ,
(n 1),
(45)
which corresponds to a logarithmic activation energy U (j ) ∼ ln(j/jc ), or to a potential law
(see section 2.5), allows one to study different levels of flux creep. An exponent n = 19 is
chosen for the simulation shown in figure 13(c) to allow studies of both the Bean critical state
(n → ∞) and creep effects; a choice n < 10 would smear the flux front markedly due to rapid
flux creep. The jc in equation (45) has the meaning of a critical current density corresponding
to a definite electric field level Ec .
The saturation value of J (y) ≈ 1.05Jc = 1.0Jc in the penetrated region depends on the
ramp rate Ḣex = dHex /dt and is caused by the inhomogeneous E(y) distribution which leads
to a spatial dependence of jc (Ec ). In figure 13(c) one has |Ḣex | = 1 in units of Ec /aµ0 .
Note that any finite ramp rate allows for flux creep, which reduces J (y) and causes a slightly
positive slope of J (y). The effect of flux creep is also seen in the magnetization curves
M(Hex ) presented for the same models in the inset of figure 13(c) for Hex cycled with three
different sweep rates |Ḣex | = 1, 10−2 and 10−4 . Note that the initial slope of the virgin curve
is independent of the sweep rate.
The qualitative agreement of the profiles J (y) and H (y) in figure 13(c) with the analytical
solution equations (37) and (41) of the Bean model in figures 13(a) and (b) is obvious. Due
to the smooth E(j ) the ideally vertical slopes of J (y) and H (y) at the flux front are smeared.
Current distributions in high-Tc superconductors
683
a)
c)
y
b)
Figure 13. (a) Measured and calculated flux density profiles across the strip shown in figure 12 for
different external fields and (b) the corresponding current density distribution according to equation
(37). The theoretical flux profiles are obtained by a least-squares fit of equations (37) and (41)
together with equation (39) with W = 200 µm, d = 250 nm and a unique jc = 2.85 × 1011 A m−2 .
(c) The theoretical model of the flux H (y) and sheet current distribution J (y) (both in units of
Jc = jc d) for a strip (a = W/2) if the electric field is taken into account. E(j ) = (j/jc )19 Ec
with jc = constant; Hex = 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 0.8, 0.9, 1 in units of Jc . The ramp rate was
dHex /dt = 1 in units of E0 /(0.5W µ0 ). The non-steplike E(J ) smears the vertical slopes of J (y)
and H (y) at the flux front as compared to the Bean model and yields a slightly varying J (y) in the
penetrated region. The inset shows the magnetization curves of this model for three sweep rates
dHex /dt = 1, 10−2 and 10−4 .
If one uses a steeper E(j ) = (j/jc )99 Ec , the ‘dynamic’ result coincides with the ‘static’
analytical results (37) and (41) within line thickness.
In order to compare magneto-optically measured flux density distributions of thin strips
with the models presented, the finite thickness (no edge singularities in the magnetic field) and
the measurement height h have to be taken into account (Knorrp et al 1994). Figure 12 shows
the magneto-optically detected flux density distribution for different applied perpendicular
fields after ZFC (Jooss et al 1996a). After calibration of the data according to equation (24),
some line profiles were taken for a defect-free region of the strip. In figure 13, the measured
magnetic flux density profiles crossways to the strip edges are compared with the calculated
profiles, using the static model of equation (37) together with equation (41). The corresponding
current densities have been obtained by a least-squares fit of the Bz -profiles to the measured
data. The relatively good agreement of measured and calculated flux density distributions
shows that equation (37) is describing well the current density in a thin strip, if the field
dependence of jc is weak and flux creep effects are small (low temperatures).
684
Ch Jooss et al
a)
b)
e)
c)
d)
Figure 14. Magneto-optically (iron garnet) detected flux distribution of a strips: (a) width
W = 800 µm and thickness d = 300 nm at T = 10 K and µ0 Hex = 40 mT. (b) The profile
of the calibrated flux density of (a). (c) and (d) show the current density after one-dimensional
inversion of the Biot–Savart law without (c) and with correction of the coupling of the in-plane field
to the magnetization of the iron garnet. (a)–(d) were taken from Johansen et al (1996). (e) Width
W = 500 µm and thickness d = 100 nm at T = 4.2 K and µ0 Hex = 16 mT. The superimposed
current flow lines and the current density profile are obtained by the two-dimensional inversion of
the Biot–Savart law which is described in section 3.
For comparison of the different magneto-optical layers and for a model-independent
determination of the current distribution, figures 14(a)–(e) depict the flux and current
distributions of YBCO strips which were measured by means of a iron garnet films.
Figure 14(a) shows a partly penetrated state of a strip with W = 0.8 mm; figures 14(b), (c) show
the calibrated flux (without taking the coupling of the in-plane component of the flux to the
magnetization into account) and the resulting current density obtained from a one-dimensional
inversion scheme Johansen et al (1996). Figure 14(d) shows the result after correction for
the in-plane coupling. Figure 14(e) depicts the flux distribution, the superimposed current
streamlines and a current density profile in a nearly fully penetrated state of a YBCO strip
(W = 0.5 mm) obtained by two-dimensional inversion of the Biot–Savart law (described in
section 3). The calibration was done according to equation (25). Since in this measurement
By Bk , the correction for the in-plane coupling could be neglected. In both cases, the
agreement with the Bean-like analytical model is convincing; no deviation from jc = constant
in the flux-penetrated areas are detectable, which could give hints as regards field dependence
of jc (B) or creep effects. We come back to jc (B) in section 6.2 and to flux creep in section 4.4.
The magnetic flux distribution of a strip with transport current was compared with the flux
distribution when applying an external field by Indenbom et al (1993a) (see also Indenbom
Current distributions in high-Tc superconductors
685
b)
a)
c)
I = 113 mA
100 µm
Figure 15. (a) Magneto-optical visualization of the flux density distribution of a bridge with a
transport current of I = 113 mA. (b) The diagram shows the geometry of the bridge. (c) The flux
profile measured across the bridge for I = 113 mA (line) and I = 44 mA (crosses) increased by a
factor of 113/44 for better comparison.
et al (1993b); MOI of large transport currents in thin-film current fault limiters is reported
in Kuhn et al (2001)). The magnetic flux density in this case is obtained, using a current
distribution somewhat different to equation (37) (Brandt and Indenbom 1993), given by

 jc
jx (y) = 2jc
(W/2)2 − Q2 1/2

arctan
π
Q2 − y 2
Q |y| W/2
|y| < Q.
(46)
Figure 15(a) shows the magneto-optically observed flux density distribution for a thin-film
bridge with a transport current of I = 113 mA. The calibrated flux density profile perpendicular
to a strip edge is plotted in figure 15(c) for I = 44 and 113 mA. Due to the larger measurement
height using an iron garnet MOL, the profiles are smoother than the profiles in figures 12 or 13.
Further magneto-optical investigations on current-carrying thin strips were performed
by Gaevski et al (1999). The burning of high-Tc bridges was visualized by the same group
(Gaevski et al 1997) with the magneto-optical technique. Further theoretical work on transport
and magnetization currents including the transport current in a magnetic field was done by
Zeldov et al (1994b). McDonald and Clem (1996) calculated theoretically the change of the
flux and current distribution for jc (B) depending on the local field (see also section 7 of this
article). Baziljevich et al (1996a) demonstrated a new method for determining jc in a thin
strip from the position of the annihilation zone formed by decreasing an external field from a
maximum value towards zero in the remanent state. Prozorov et al (1993) investigated the flux
exit and remagnetization of thin strips in a decreasing external magnetic field. Vlasko-Vlasov
et al (1993a) imaged the planar field component of a YBCO thin strip by means of iron garnet
indicators. For discs and thick samples, see Yao and Aruna (2000). For experimental results
on discs, see also section 6.2.
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Ch Jooss et al
4.2. Arbitrarily shaped films—critical state
In order to proceed step by step to an understanding of the full three-dimensional problem
of flux penetration into a type-II superconductor we continue in the following with a
consideration of thin superconducting films of arbitrary shape. Here, a two-dimensional
j (x, y) = ex jx (x, y) + ey jy (x, y) is realized for any field direction within the film’s x–y
plane because the current component normal to the film plane is negligibly small; jz = 0
for d < λ. In addition, for d < λ or d < aF LL at large magnetic fields (aF LL denoting the
lattice constant of the flux line lattice), the z-dependence of j can be completely neglected.
For experimental investigations, different geometries such as rectangles, squares, triangles and
polygons can be easily realized by thin-film patterning methods.
For simplicity, we first consider in this section the critical (fully penetrated) state, where
j = jc = constant over the whole sample. The more complex partly penetrated state in
arbitrarily shaped films, where the magnitude j = |j | as well as the orientation ĵ = j /j is
varying, is treated in section 4.3.
In the stationary state, where the density ns of the superconducting charge carriers is
constant in time (n˙s = 0), one has
∇·j =0
in I,
j·n=0
on .
(47)
Here, ∇ is the 2D divergence operator, I the area of the superconductor, the boundary of I
and n a vector normal to . Equation (47) is obviously valid in magnetization experiments on
isolated superconductors, where no external current is fed into the specimen and the current
streamlines are closed within the superconductor. In addition, it applies also to superconductors
with transport current, when the contact area is excluded, where a normal current is transformed
into a supercurrent (or vice versa) and n˙s = 0. Equation (47) reflects that all normal components
of the supercurrent (j · n)n are vanishing at all surfaces and the supercurrent has to follow
the film edges.
As a consequence of this boundary condition, a vector field j with constant modulus splits
up into domains with uniform parallel current flow (van den Berg 1986). As discussed in
the review by Campbell and Evetts (1972), these domains with different j -orientations and
different directions of the magnetic flux gradients are separated by sharp bends in ĵ , forming
discontinuity lines (d + -lines) (Campbell and Evetts 1972, Schuster et al 1994a, Jooss et al
1998a). The d + -lines appear at corners and recesses at the edges of the superconductor or
may also have their origin at holes or large normal precipitates inside the samples. Note that
domains with uniform parallel current flow may be separated also by regions with continuous
changes in the j -orientations, e.g. at concave corners or holes (see section 4.5). Note that
discontinuity lines had already been observed in metallic superconductors, by De Sorbo and
Healy (1964).
According to the Ampère or the Biot–Savart law, the mesoscopic magnetic flux distribution
must be self-consistent with the mesoscopic current density distribution. This applies not only
in the framework of mesoscopic electrodynamics: also from the microscopic viewpoint, which
considers forces acting on vortices, the self-consistency must be conserved. This means that the
directions of the driving and counteracting pinning forces in equation (11) are determined by the
boundary conditions in equation (47). Another aspect of the self-consistency concerns the local
magnetic field outside the superconducting sample: it is given by the superposition of the stray
fields of the supercurrents with the applied external field and therefore is strongly dependent
on the shape of the sample. In general, the stray field of the current flowing around a convex
corner is smaller than that at a straight edge or a concave corner. The local value of the magnetic
field at a sample edge directly influences the penetration depth of the flux front in the partly
penetrated state and thus the area in which the critical current is flowing. Further complications
Current distributions in high-Tc superconductors
687
Figure 16. Current streamlines (top) and the normal component of the magnetic field in the critical
state as a contour plot (middle) in a thin type-II superconductor of square (a) and rectangular (b)
shape according to Brandt (1995c). Bottom: greyscale images of the magnetic flux distribution
of square (c) and rectangular (d) shape YBa2 Cu3 O7 thin films in the fully penetrated state. (e)
The remanent state of the rectangle after full flux penetration. Sample geometries: square with
d = 700 nm and a = 1 mm; rectangle with d = 130 nm and a/b = 0.5 (b = 1 mm); T = 4.2 K.
arise for the current distribution if dependences of jc on the local magnetic B(x, y) or electric
field E(x, y) are taken into account (see sections 4.4 and 6).
For simplicity and to be more specific, the following discussion will be focused first on
square and rectangular superconducting thin films with lateral sizes a and b. The generalization
to the critical state for arbitrary shapes will then follow immediately. For superconductors with
rectangular cross section it follows from the conditions given above that the current streamlines
have sharp bends (figures 16 and 17). The current distribution shows a pattern of current
domains of uniform parallel current flow which are separated by current domain boundaries.
In the framework of the Bean model with stepwise current–electric field characteristics, the
bending of the current is discontinuous and thus the size of the current domain boundaries
becomes infinitesimally small. In this case they form so-called discontinuity lines (for the
case of non-linear E(j ) characteristics, see the next section). One distinguishes two types of
d-line (Schuster et al 1994a): at d + -lines the orientation of jc changes discontinuously but
the magnitude of jc remains the same. At d − -lines the magnitude of jc changes, e.g., at the
specimen surface or at inner boundaries where regions of different jc meet (see section 6).
688
Ch Jooss et al
Figure 17. Top: the measured flux distribution (after calibration) of a square-shaped YBa2 Cu3 O7
film (d = 200 nm, a = 1 mm) similar to the remanent state. After application of a maximum
external field of 400 mT, µ0 Hex was reduced to 56 mT. T = 4.2 K, MOL: (Lu, Bi)-doped iron
garnet. Bottom: the corresponding current density distribution, where the current streamlines are
superimposed on the MO image and the current density profiles of jx and jy are given at the
positions of the broken lines. The current density is obtained from the inverted Biot–Savart law
according to equation (36).
In isotropic rectangular superconductors, the d + -lines run along the bisection lines starting
from the sample corners and on a section of the middle line parallel to the longer side as shown
in figures 16 and 17. Characteristic features of the d + - and d − -lines are:
• Whereas the d − -lines occur at internal and external boundaries of the sample, the d + -lines
form in homogeneous regions and are determined by the shape of the sample.
Current distributions in high-Tc superconductors
689
b
a
d
c
β
b
w
f
e
g
h
L2
w
L1
i
y
R
L1
x
Hm > H*
j
Figure 18. Current streamlines and the calculated distribution of the normal component of the
magnetic flux density Bz for different thin-film geometries in the (fully penetrated) critical state.
(a), (b) Triangular sample, (c), (d) irregular polygon-shaped film, (e), (f ) square-shaped film with
isotropic and (g), (h) with anisotropic current (anisotropy ration 0.5) and (i), (j) a rectangular sample
with a semicircular defect with radius R at one edge (Forkl and Kronmüller 1995).
• Flux lines cannot cross the d + -lines since during increase or decrease of the applied
magnetic field the flux motion is directed towards or away from the d + -lines, respectively.
In contrast, the d − -lines can be crossed, e.g., when flux lines penetrate from the surface.
When the current does not flow parallel to the d − -line, a strong flux motion is directed
along the d − -line.
• The electric field E is largest at the d − -lines, whereas E = 0 at the d + -lines.
• In the framework of the Bean model with jc = constant, the d + - and d − -lines do not
change their position during lowering or reversing of the external magnetic field, although
the magneto-optically detected intensities of the d + - and d − -lines are reversed in the
remanent state. For a magnetic field dependence of the position of the d + -lines, see
section 8.2.
In thin type-II superconductors (d a, b), these d + - and d − -lines are clearly seen because
of the logarithmic infinity of Bz at the sample surface (Schuster et al 1994a).
The rectangular current pattern of the critical state applies to superconductors of arbitrary
thickness d. The resulting magnetic field, however, depends on d. If the specimen’s cross
section is |x| a, |y| b, then in longitudinal geometry (for large specimens with length d
along Hex ẑ ) the magnetic field in the Bean critical state inside the superconductor is
Hz (x, y) = Hex − jc max(a − |x|, b − |y|),
(48)
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Ch Jooss et al
Figure 19. The measured flux distribution of a triangle-shaped YBa2 Cu3 O7 film after ZFC to
4.2 K for external magnetic fields of µ0 Hex = 20 mT (a), 32 mT (b), 64 mT (c) and 32 mT after
application of a maximum field of 300 mT (d). In (e) the current streamlines are superimposed on
(b) and a current density profile of jy (x) is shown along the broken line. MOL: iron garnet. The
black spot in (d) is a defect in the MOL.
where max(· · ·) denotes the maximum value, and outside the superconductor one has
Hz (x, y) = Hex . This well known field profile has constant slopes and looks like a sandpile. In this longitudinal geometry the lines Hz (x, y) = constant coincide with the current
streamlines. For intermediate specimen thickness d, the magnetic field of the rectangular
current pattern takes a more complicated form which depends on d in a non-trivial way; see
the comprehensive paper by Forkl and Kronmüller (1994). For thin specimens, d a b,
the magnetic field again becomes independent of the thickness. From the Biot–Savart law one
finds the following perpendicular field component in the specimen plane z = 0 inside and
outside the superconductor (Schuster et al 1995b, Brandt 1995a):
jc f (px, qy)
Hz (x, y) = Hex +
4π p,q=±1
√
(49)
√
(P + y − b)(y − b + a)(P + x − a) x 2P + a + b − x − y
f (x, y) = 2 ln √
+ ln (y − b)(Q + y − b + a)(x − a)(Q + x) 2Q − a + b − x − y
with P = [(a − x)2 + (b − y)2 ]1/2 , Q = [x 2 + (b − a − y)2 ]1/2 . This result is different to the
field pattern obtained by the sand model (Campbell and Evetts 1972) which is valid for thick
samples.
For arbitrary sample thickness, the generalization of the sand model applies to the local
magnetization g(x, y) which is given by Brandt (1992a):
j (x, y) = −ẑ × ∇g(x, y) = ∇ × ẑ g(x, y).
(50)
The local magnetization g has also the meaning of a density of tiny current loops. Applying
the sand model to this quantity, one can determine g(x, y) from the following simple relation
(Prigozhin 1998):
g(x, y) = jc dist(x, y, ),
(51)
Current distributions in high-Tc superconductors
691
which determines completely the current distribution in the fully penetrated critical state
jc = constant. Here ‘dist’ is the distance function giving the distance of a position (x, y)
inside the superconductor from the sample border . The equipotential lines of g are the
current streamlines of jc . Note that for thick samples one has g(x, y) ∝ H (x, y).
Based on the observation that the current distribution in the critical state splits up
into domains of uniform parallel current flow, Forkl and Kronmüller (1995) constructed
the current and magnetic distributions for various film geometries by using a cake-shaped
basic brick (see also Forkl et al (1996b, 1997)). In this model each brick represents
a domain with parallel current flow, where the current density has a constant magnitude
and angle. Since the magnetic stray field of the current in a brick can be calculated
analytically, the flux density distribution even of films of complex shapes can be calculated as
a superposition of the stray field of all bricks. Figure 18 shows the model of the current and the
corresponding normal component of the flux distribution in differently shaped films. Figures 17
and 19 give experimental results for a square-shaped and a triangle-shaped YBa2 Cu3 O7 film,
respectively.
4.3. Thin films—partly penetrated state
Up to now, we have only considered the fully penetrated state where j = jc over the whole
superconductor. In contrast to the problem of a one-dimensional current distribution in an
infinitely extended strip or in a disc, where the problem of the shielding current in the partly
penetrated state has been solved analytically (Brandt et al (1993), Mikheenko and Kuzlovlev
(1993), Zhu et al (1993); for an elliptic film see Mikitik and Brandt (1999)), no analytical
solution for the distribution of the shielding current has been found for the two-dimensional
problem up to now.
In this section we present a short outline of Brandt’s theory of the general equation of
motion for the sheet current J (x, y, t) = j d in a thin planar conductor or superconductor of
thickness d and arbitrary shape in a time-dependent perpendicular applied field ẑ Hex (t). A
detailed derivation and further applications to eddy currents, linear ac response and flux creep
are given elsewhere (Brandt 1995b). The material will be characterized by B = µ0 H and by
a resistivity ρ = E/j or sheet resistivity ρd = E/J = ρ/d, which may be non-linear, e.g. a
power law ρ(j ) = ρc (j/jc )n−1 (Rhyner 1993), or linear, complex and frequency dependent,
ρ = ρac (ω) = ρ + iρ . In general, the non-linear ρ may depend on B via jc (B) and n(B), and
the linear ρ via the factor B/Bc2 in the flux flow resistivity ρF F ≈ ρn B/Bc2 , where ρn is the
normal resistivity. In the theory the sheet resistivity may depend on the position, ρd = ρd (x, y),
either directly in a non-uniform specimen, or indirectly via j (x, y) and B(x, y).
The equations for the sheet current J and for the perpendicular induction component
Bz = µ0 Hz in the specimen are obtained as follows. Firstly, one has to express the sheet
current by a scalar function G(x, y) = g(x, y)d as
J (x, y) = −ẑ × ∇G(x, y) = ∇ × ẑ G(x, y).
(52)
Next one determines the integral kernel Q(r , r ) (r = (x, y)) which relates the
perpendicular field Hz (x, y) in the specimen plane z = 0 to the local sheet magnetization
G(x , y ) by
Hz (r ) − Hex =
Q(r , r )G(r ) d2 r ,
(53)
A
G(r ) =
Q−1 (r , r )[Hz (r ) − Hex ] d2 r ,
(54)
A
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Ch Jooss et al
Figure 20. Left image: the calculated current pattern (left column) and perpendicular field Hz
(middle column) distribution of a square sample for three different steps of flux penetration. Right
column: magneto-optically detected flux distributions in a square YBCO thin film after ZFC to
T = 50 K at µ0 Hex = 54 mT (top), 92 mT (middle) and 151 mT (bottom). The flux distributions
were detected using an iron garnet indicator. The black spots are defects in the indicator film. The
film thickness is d = 800 nm. In the calculations the unit of the magnetic field (one fit parameter)
is chosen such that best agreement is found with the observed flux distributions. Right image: as
left image but for decreasing magnetic field. Top: µ0 Hex = 88 mT; middle: 30 mT; bottom: 0 mT
(remanent state).
where the integrals are taken over the specimen area A and Q−1 is the inverse kernel. Due
to problems with singularities, the finding of a well behaved integral kernel Q is a non-trivial
task (Brandt 1992a, 1995c, Schuster et al 1995a). Noting that the field of a tiny current
loop (or magnetic dipole) of unit strength located at x = y = z = 0 with axis along z is
Hz (x, y, z) = (1/4π)(2z2 − x 2 − y 2 )/(x 2 + y 2 + z2 )5/2 , one obtains
Q(r , r ) =
2
2z2 − rxy
1
= lim
Kg (r − r )
lim 2
2 )5/2
z,z →0
4π z→0 (z + rxy
(55)
2
with rxy
= (x − x )2 + (y − y )2 and Kg defined according to equation (32).
From the integral kernel Q the inverse kernel Q−1 may be obtained by Fourier
transformation (Brandt 1995b, 1995c) or by introducing a grid with positions ri = (xi , yi ),
= Q(ri , rj )wj . The
weights wi , the tables Hi = Hz (ri ), Gi = G(ri ) and the matrix Qij
integrals (53) and (54) are then approximated by the sums Hi =
j Qij Gj and Gi =
−1
−1
j Qij Hj where Qij is the matrix inverse of Qij (Brandt 1994b, 1994c, Xing et al 1994).
As the last step, the equation of motion for G(x, y, t) is obtained from the (3D) induction
law ∇ × E = −Ḃ and from the material laws B = µ0 H and E = ρ j valid inside the sample
where j = J /d = −ẑ × (∇G)/d. In order to obtain an equation of motion for G(x, y, t), it is
important to note that the required z-component Ḃz = ẑ Ḃ = −ẑ (∇ × E ) = −(ẑ × ∇)E =
−x ∂ˆ E /∂y + y ∂ˆ E /∂x does not depend on the (unknown) derivative ∂ E /∂z. With the sheet
resistivity ρd = ρ/d one may write for inside the sample E = ρ j = ρd J = −ρd ẑ × ∇G and
thus Ḃz = (ẑ × ∇)(ρd ẑ × ∇G) = ∇ · (ρd ∇G). Inserting this into equation (54) one obtains
Current distributions in high-Tc superconductors
the equation of motion for G(x, y, t) in the form
Ġ(r , t) = Q−1 (r , r )[f (r , t) − Ḣex (t)] d2 r ,
693
(56)
f (r , t) = ∇ · (ρd ∇G)/µ0 .
This general equation, which applies also when ρd = ρ/d depends on r , j and Hz , is
easily integrated over time on a computer. Since ρ/µ0 has the meaning of a diffusivity,
equation (56) describes non-local (and in general non-linear) diffusion of the local sheet
magnetization G(x, y, t). When the resistivity is anisotropic, equation (56) still applies
but with modified f (r , t). For example, if Ex = ρxx jx and Ey = ρyy jy , one has
f (r , t) = ∇x [(ρyy /d)∇x G] + ∇y [(ρxx /d)∇y G].
By solving these equations numerically, the current, magnetic and electric field distribution
have been obtained for squares and rectangles (Brandt 1995b, Schuster et al 1995a), crossshaped samples and samples intended to be rectangular (Schuster et al 1996a). Figure 20
shows the development of the current streamlines and the magnetic flux density distribution
for a square-shaped thin film in increasing and decreasing external field.
In contrast to the case for the parallel geometry, where the shielding currents flow only in
the penetrated regions to ensure Bz = 0 in the Meissner area, in the perpendicular geometry
shown in figure 20 the shielding currents are flowing throughout the entire sample. Comparing
this partly penetrated state with the fully penetrated state in figures 16 and 17, one finds that
during the magnetization process the shielding currents change their direction as well as their
magnitude until the critical value jc is reached. The current streamlines turn monotonically
from the initial concave shape during the magnetization process into straight lines flowing
parallel to the film edges. This can be seen also in figure 22, where the measured flux
density and the calculated current density distributions are shown. The current density is
obtained independently of the model by 2D inversion of the Biot–Savart law. During virgin
flux penetration, the streamlines in a square never become convex as was supposed by Brüll
et al (1991).
We find that the regions with j = jc grow from the middle of the sample edges. The
current-generated magnetic field, which is oriented with the external magnetic field outside
the sample, superimposes on the external magnetic field such that a large field enhancement
is observed at the sample edges. This overshoot is maximum at the middle of the sample
edges (see figure 21). As a result, the magnetic flux penetrates the samples in a cushion-like
pattern (Rowe et al 1971, Brüll et al 1991, Vlasko-Vlasov et al 1992, Schuster et al 1992b),
i.e., from the middle of the edges rather than from the sample corners as could be expected
naively. The flux-free dark zone in the sample centre is the Meissner phase. In figure 20 the
neutral line, where the flux profile intersects the value of the external field, is emphasized as
a bold line in the contour plots of the magnetic field distribution. The numerically calculated
flux distributions nicely agree not only with the magneto-optical images in the right column
in figure 20. The same is also true for the current patterns in figures 20 and 22. The critical
state coincides with the analytical solution shown in figure 18. The above-mentioned d + -lines
are visible in the magneto-optical images of the critical state as dark lines running along the
diagonals.
In decreasing external field the flux lines exit the sample due to the reversal of the driving
Lorentz force. This is shown in figure 20 for µ0 Hex = 88, 30 and 0 mT (remanent state) and
also in figure 22(d). The reversal is caused by the flip of the current from +jc to −jc , which
occurs near the sample edges. As soon as the direction of the change of Hex is reversed, the
current density falls below jc everywhere; relaxation will therefore stop effectively (Brandt
1992b, 1992c).
694
Ch Jooss et al
Figure 21. Top: measured flux distribution (after calibration) of a square-shaped YBa2 Cu3 O7
film (d = 200 nm, a = 1 mm) after ZFC to T = 4.2 K and application of µ0 Hex = 32 mT.
MOL: (Lu, Bi)-doped iron garnet with Bk = 220 mT. Bottom: the corresponding current density
distribution obtained by inversion of the Biot–Savart law: the current streamlines are superimposed
on the MO image and the current density profiles of jx and jy are given at the positions of the broken
lines. Note that the finite current outside the sample and some of the current increase towards the
sample’s edges are a result of the calibration errors (inhomogeneous illumination) described in
section 3.
The turning of the current direction is seen from the loops in the current pattern in the
left column in figure 20 (right image). At the zero-line of the current the current-generated
magnetic fields of both domains of opposite current direction are oriented parallel to each other.
This leads to a maximum in the magnetic field distribution there; see the loops in the contour
plots of the magnetic field distribution depicted in the middle column and the magneto-optical
Current distributions in high-Tc superconductors
695
Figure 22. Development of the flux and current distribution in a magnetization experiment for
a square-shaped film (d = 200 nm, a = 1 mm) after ZFC to T = 4.2 K and application of
µ0 Hex = 32 mT (a), 56 mT (b), 80 mT (c) and reduction to 56 mT from µ0 Hmax = 400 mT (d).
images in the right column in figure 20. The exiting flux lines cause a dark zone of reduced
flux line density which spreads from the edges into the sample. On reaching the remanent
state (Hex = 0, bottom row in figure 20) flux lines with opposite sign, as compared to the
pinned ones, start to penetrate the sample and partly annihilate with the pinned flux lines
(Szymczak et al 1990, Schuster et al 1991, 1992a, 1992b, 1992d, 1992e, Forkl et al 1991,
Vlasko-Vlasov et al 1993b). The boundary between regions containing flux lines of opposite
polarity is indicated by the bold line Hex = 0 in the middle plot in the bottom row in figure 20.
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4.4. Finite electric fields
The principal question is how the critical state of a flat HTS sample is modified if we take
flux creep and the related electric field distribution into account—which is highly non-uniform
and depends on the samples geometry. In the Bean model which was considered in an earlier
section, the real non-linear E(j ) curve is approximated by a stepwise dependence, E = 0
for j jc and E → ∞ for j > jc , which is highly non-physical. Although exponentially
small at j < jc , even in the flux creep state the non-zero electric field can play an important
role in the spatial distribution of the current density. More important is to consider the highly
inhomogeneous electric field distribution in states with a high electric field level. In this case,
the current density in a sample displays spatial variations because different areas lie at different
points on the E(j ) curve.
Firstly, we discuss the geometry-dependent distribution of the electric field E(x, y) during
flux flow when the external magnetic field is swept. During flux penetration the change of the
magnetic flux density with time induces an electric field according to ∇ × E = −Ḃ . The
electric field is caused by the losses during flux movement which leads to an electric resistivity
ρ = E /j . In longitudinal geometry, where the original Bean model applies, one has in the
regions with straight parallel current streamlines, x̂ (Brandt 1995c),
E = E(x, y)x̂,
E(x, y) ≈ µ0 Ḣex y + f (x).
(57)
However, for realistic geometries with finite thickness or for thin films, equation (57) is not
valid and the electric field can be obtained from equation (54) using a non-linear current–
electric field relation E(j ) = Ec (j/jc )n . As an example, the electric field pattern of a square
during flux penetration in a partly penetrated state is depicted in figure 23(a). It exhibits a
maximum value at the middle of the square edges. In contrast to the case for the E(x, y) pattern
during flux creep (Ḣex = 0) which is depicted in figure 23(b), during flux flow E increases
monotonically towards the sample edges. Since the local value of E(x, y) = Bv depends on
the local flux density B(x, y) and the local vortex velocity v, the maximum of E is related to
the maximum of B at the middle of the sample edges. For isotropic resistivity ρ, the electric
field is always parallel to the current density E j .
In comparison to that in the flux flow state, E(x, y) in the flux creep state is as depicted
in figure 23(b). The depicted E-pattern does not change over many decades of the creep time
t; however, the amplitude decreases as E ∝ t −n/(n−1) (Brandt 1995c) (for arbitrary shape, see
(Brandt 1996b)). The highest electric field level appears at the middle of the square edges,
whereas E exhibits minima at the d + -lines and at the singular point at the sample centre, where
the d + -lines are intersecting.
From the electric field distribution in figure 23 it follows that different locations in the
sample lie at different positions on the E(j ) curve. Consequently, even in the critical state, the
magnitude of the current density may become a function of the position within the x–y plane of
the sample. Indeed, the current densities in figure 23 exhibit a tendency for the largest current
density to appear at the position of the largest electric field at the middle of the sample edges
in increasing field and a minimum in decreasing field. This indicates that the critical current
density jc at the sample border is measured at a higher field criteria Ec than the jc towards
the sample centre (see also Yao et al (2000) for thick samples). However, the calibration
errors described in section 3.3 could produce a similar effect, which cannot be separated off
completely at present.
In contrast to the analysis of the jc (E) problem, the direct experimental observation of the
electric field distribution is possible, by time-resolved magneto-optical imaging (Jooss et al
2001b). According to equation (12), due to the inhomogeneous electric field distribution the
time decay of the current density becomes a function of the sample position, where the decay
Current distributions in high-Tc superconductors
697
Figure 23. (a) The calculated magnitude |E (x, y)| of the electric field in a flux flow state during
flux penetration (Ḣex = 0) into square-shaped thin-film superconductors. (b) The pattern of the
electric field E(x, y) during flux creep for a fully penetrated state of a square-shaped thin film. For
the calculation, an exponent n = 39 of the E(j ) curve was used (Brandt 1995c).
Figure 24. A greyscale image of the changes in the absolute value of the current density distribution.
The image represents a subtraction of the initial current distribution at t ≈ 1 s from the one 300 s
later. Dark and bright parts therefore refer to areas where the current density was decreased or
increased during flux creep, respectively. Note that |j (x, y)| is directly proportional to the electric
field distribution E(x, y).
rate is largest at the sample edges and has a minimum at the d + -lines. For thin films, one
can derive from ∇ × E = −Ḃ that the time derivative of the local current density is directly
proportional to the local electric field value E = d2 µ0 ∂j/∂t (Jooss et al 2001b).
Figure 24 shows the change of the absolute value of the current density from an initial
state t ≈ 1 s after applying an external magnetic field to a relaxed state at t = 300 s. At
those parts of the sample where magnetic flux has penetrated at t ≈ 1 s, the current density
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Ch Jooss et al
b)
a)
200 µ m
Figure 25. Superposition of the magnetic flux density distribution and the current streamlines in
the region of a current domain boundary for an almost fully penetrated state of a square-shaped
YBCO film (d = 200 nm) at T = 4.2 K (a). In (b) the absolute value |j |(x, y) of the current
density is visualized as a greyscale image for the same conditions as in
√ (a). The current domain
boundary is visible as an apparent drop of the jc -value by a factor of ≈ 2 for geometrical reasons
(see the text).
is decreased during the relaxation process. In agreement with the electric field distribution in
figure 23, the relaxation is strongest at positions of largest electric field at the sample edges.
A minimum of the current relaxation occurs at the initial position of the flux front (where j
changes sign) and remarkably also at the current domain boundaries. This proves that the
electric field during flux creep is indeed small at the d + -lines. One observes an increase of
j in initially flux-free parts of the square, where due to the creep-induced propagation of the
flux the shielding current turns into a critical current. In this area, the electric field is directed
oppositely to the current. As shown by Warthmann et al (2000), the time decay of the flux and
current distribution is also influenced by a magnetic field dependence and spatial variations of
the activation energy. For local analysis of relaxation see also Abulafia et al (1995).
Another consequence of a finite electric field is that the discontinuity of the d + -lines of the
Bean model is removed and one observes current domain boundaries, where the current is bent
on a small but finite length scale (Gurevich and McDonald 1998). According to Gurevich and
Friesen (2000) the stepwise change of the current orientation θ(x, y) = π/4[sgn(x − y) − 1] at
the Bean d + -lines develops into a smooth distribution θ(x, y) = π/4 + δ(x, y), where δ(x, y)
is a power-law function of the coordinates near the diagonal which also strongly depends on
the exponent n of the E(j ) curve. If one defines a width db of the current domain boundary as
a region where δ < δ0 (δ0 denoting an arbitrary cut-off angle limiting the domain boundary),
Gurevich and Friesen (2000) found that, for transport currents, db = δ0 (3/n + δ02 )l linearly
increases with the distance l from the sample corner along the diagonal.
It is important to notice that, in the Bean limit n → ∞ also, there is still a finite size db of
the current domain wall. The discontinuous d + -line configuration of the Bean model satisfies
the critical state assumption j = jc = constant, with a continuity of the normal component
of j and a jump of the tangential component of j at the d + -line. A jump of j would result
in a discontinuity of the tangential component of E which violates the Maxwell equation
∇ × E = −Ḃ ≈ 0 in the (slowly relaxing) steady state. Consequently the Bean discontinuity
lines cannot exist in a steady state if any (exponentially small) electric field E(j ) is taken into
account.
Figure 25 shows a part of the current distribution of a square-shaped YBCO film containing
a current domain boundary at the diagonal of the sample. The current streamlines clearly show
a continuous change in their orientation on a length scale of ≈45 µm which is much larger
than the spatial resolution of the measurement (3 µm). The greyscale image of the absolute
Current distributions in high-Tc superconductors
699
value |j (x, y)| of the current density in figure 25 reveals a minimum of |j (x, y)| at the domain
wall which has mainly geometrical
reasons: without discontinuity, the cross section for the
√
total current is increased by ≈ 2 along the diagonal of the square.
4.5. Convex corners, crooked edges and holes
As already pointed out in section 4.2, domains of uniform parallel currents of different orientations are not always separated by sharp boundaries (d + -lines), where the current orientation
is turned discontinuously. Due to small but finite electric fields the discontinuity is removed
and one observes current domain walls of finite size at the bisection lines of convex corners. A
more complex situation arises if we consider a superconductor with convex corners, non-perfect
edges or holes within the sample. In this case, the direction parallel to the sample edges is no
longer defined in a certain region of the superconductor behind the corner. As an example, we
consider in the following a cross-shaped sample, possessing eight concave and four convex corners. For the current flow in the region near the convex corners two alternatives are conceivable,
at first thought—by extrapolating the current streamlines parallel to the sample edges (Schuster
et al 1996a). For the fully penetrated state these two alternatives are depicted in figure 26.
In the first possibility shown in the upper plot, the current streamlines are linearly extended
beyond the corner and meet at an angle of 90◦ such that a d + -line must be formed along the
bisection line. In the second possibility the streamlines run on concentric circles around the
corner as shown in figure 26(b). In this case no d + -line is formed.
Figure 26 shows the calculated current pattern, a contour map of the calculated flux density
distribution and magneto-optically visualized flux distributions for the cross-shaped sample.
The calculations have been performed by using equations (53) and (54) (Schuster et al 1996a).
Both the calculation and the measurement imply that the second possibility of figure 26 is
realized. In contrast to the case for convex corners, there is no d + -line at the convex corners
and the current streamlines change their direction continuously. Note that this behaviour is
also predicted by equation (51).
Similar concentric current patterns also arise in the vicinity of notches in the specimen
edge or holes (e.g. due to large normal-conducting defects) in the superconducting specimen.
Campbell and Evetts (1972) considered the current distribution around a cylindrical cavity
which is located near to the edge of a thick superconductor for a magnetic field parallel to
the cavity axis. They demonstrated that because jc = 0 in the cavity and div j = 0 around
the cavity, the current has to follow the circular shape of the cavity, thus forming concentric
current flow lines. The transformation from the straight current flow parallel to the specimen
edge to the concentric flow around the cavity takes place at sharp parabolic d + -lines.
As pointed out by Schuster et al (1994a) the effect of the modified current flow on the flux
pattern near to holes and notches is strongly enhanced in thin films. The thin-film geometry
leads to larger stray fields at all kinds of d-line. Consequently, the sharp peaks in the magnetic
flux distribution at d + - and d − -lines assist the observation of such effects by magneto-optical
measurements. Figure 27 shows the flux and current distribution around a macroscopic defect
exhibiting poorer superconducting properties with a diameter of ≈30 µm near an edge of a
YBCO square. Figure 27(c) depicts the current flow lines assuming a spatially constant current
density outside the defect (Bean model) and equation (47). The bending points which form
the current domain boundary are given by the intersection of the circular currents around the
cavity with the current lines parallel to the sample edges. They form a parabola (Schuster et al
1994a, Jooss et al 1998a)
y=
1 2
x − Rdef ,
Rdef
(58)
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Ch Jooss et al
Figure 26. (a), (b) Sketches of two alternatives for the current streamlines in a cross-shaped
sample in the critical state. In (a) the streamlines meet at 90◦ at the concave corner and a d + -line
is formed. This possibility is not realized in superconductors. The streamlines in (b) running on
concentric circles are realized in experiment. (c) The calculated current pattern (left plot) and
perpendicular magnetic flux density Bz (right) of a cross-shaped sample. (d) Magneto-optically
observed flux density in a cross-shaped DBCO single crystal patterned by laser cutting at T = 20 K
and µ0 Hex = 82 mT. The black spot is a defect in the iron garnet film used. (e), (f ) Flux distribution
and superimposed current streamlines obtained by inversion of the Biot–Savart law for a patterned
YBCO film (d = 130 nm) at 4.2 K and µ0 Hex = 16 and 48 mT, respectively.
where Rdef denotes the radius of the defect modelled as a cylindrical cavity with jc = 0.
Note that equation (20) in Jooss et al (1998a) is wrong and must be corrected by a factor
of 2. However, the defect sizes estimated in this article have been obtained using the correct
equation.
The agreement between the Bean model sketched in figure 27(c) and the modelindependent result of figure 27(a) is quite good. In contrast to the case for the d + -line of
the Bean model, one observes a continuous bending of the currents which is related to a small
drop in |j |(x, y) visible in figure 27(b); however, the shape of the domain boundary agrees
well with the parabolic shape predicted by the Bean model. There is only a slight deviation
from a parabolic shape of the d + -lines due to the dependence of jc (B) on the local magnetic
flux density B. The increase of jc from the sample edge to the flux front due to the reduced
local Bz leads to an additional widening of the parabola branches.
If macroscopic defects are located at a larger distance from the sample’s edge, current
patterns with concentric circles on both sides of the holes may be formed (Jooss et al 1998a,
Current distributions in high-Tc superconductors
a)
c)
701
b)
|J|
|j(x)|
Figure 27. The modification of flux and current distribution at a macroscopic defect near the
edge of a square-shaped YBCO film with thickness of 300 nm at µ0 Hex = 48 mT and T = 5 K.
(a) The magnetic flux distribution and superimposed current flow lines. (b) The absolute value of
the current density for the same area. The macroscopic defect with diameter of ≈30 µm is visible
as a positive peak in Bz and a sharp minimum in |j |. Note that jc (B) is increased towards the flux
front due to its dependence on the local magnetic field. (c) A sketch of the current flow lines near a
cylindrical cavity (black circle) in a superconducting strip or square in the framework of the Bean
model (jc = constant in the flux-filled area). The straight thick line represents the sample border.
Baziljevich et al 1996b). This leads to a double-peak feature in the flux distribution at the
defect position, with a local maximum and a local minimum in Bz (x, y). Further, it is shown in
this article that in thin discs the current distribution and the shape of the d + -lines around large
defects are completely different. The shape of the d + -lines is elliptical instead of parabolic
which is, however, a purely geometric effect. Some more magneto-optical visualizations of
macroscopic defects are reported in Koblischka et al (1995), Koblischka (1996) and Jooss et al
(1998b) for scratches and notches in YBCO thin films and by Kuhn et al (1999a) for defects
in large-area YBCO thin films.
After having considered the concentric current patterns at convex corners and holes
together with the corresponding Bz (x, y), we finally come back to the consequences of
concentric current patterns for the flux movement, electric field distributions and dissipation
processes. As already pointed out at the end of the previous section, the electric field is always
parallel to the current density, E j , for isotropic resistivity ρ. This is also valid when the
streamlines form concentric circles; E = ρ j . The magnitude of the electric field due to flux
motion during sweeping the external field by Ḃex is approximately given by (Schuster et al
1996a)
2
rp (ϕ)
1
E (r, ϕ) ≈ Ḃex
− r ϕ̂.
(59)
2
r
Here r and ϕ are polar coordinates centred at the origin of the concentric current pattern,
ϕ̂ = (−y x̂ + x ŷ )/r is a unit vector along ϕ and rp (ϕ) has the meaning of a radial penetration
depth of the flux front. The mid-point of the circles coincides with the location of a convex
corner or with the centre of a macroscopic defect in the superconductor. It is the location of a
1/r peak in the electric field being created by the moving vortices. As an example, figure 28
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Ch Jooss et al
Figure 28. (a) A sketch of the concentric current streamlines and the flux movement at a convex
corner. The field lines of E are parallel to j . (b) The calculated magnitude |E (x, y)| of the electric
field during flux penetration into a cross-shaped superconductor for a partly penetrated state. The
split in the peaks is due to the calculation on a grid in the x–y plane.
presents the electric field distribution for the cross-shaped sample during flux penetration in a
partly penetrated state.
The reason for this peak in the mesoscopic E (x, y) on the microscopic level is that all
vortex lines penetrating into the regions of concentric current patterns have to pass through a
small area around r = 0 (or through r Rdef for macroscopic defects). Since the flux line
motion is always directed perpendicular to the current flow, the magnetic flux cannot fill up
the region of the concentric flow without moving though the singular point r = 0. The peak in
the electric field is related to a maximum of the locally dissipated power p = J · E . Therefore
these results are particularly important for the performance of superconductors in high-field
or high-current applications, where an unfavourable geometry with convex corners, a high
density of holes (e.g. normal-conducting defects) or scratches may lead to thermal instabilities
and enhanced losses in ac applications.
4.6. Superconductors of finite thickness
In the earlier sections we have considered the magnetic flux penetration and the corresponding
current distribution either in the limit of an external magnetic field parallel to an infinitely
extended cylinder (original Bean model) or in the limit of a magnetic field being applied
perpendicular to the plane of a thin film. In both limits the supercurrent density does not
depend on the coordinate z parallel to the applied external field Hex . In this section we will
proceed now to the problem of three-dimensional flux penetration in samples of finite thickness
d; λ < d W . There are two problems which are important to consider for such samples: the
first problem concerns the supercurrent distribution in the x–y plane, when an external field is
applied in the normal direction. The second problem is the flux penetration from the surfaces
oriented perpendicular to Hex . This problem is strongly linked to the question of the origin of
the observed shielding currents j jc in the flux-free areas of samples in the perpendicular
limit with d W .
The normal magnetic flux density Bz is usually measured at a constant measurement
height h above the surface of the superconductor. Consequently, the investigation of the z-
Current distributions in high-Tc superconductors
703
b)
a)
c)
Figure 29. Left: a greyscale image of the magnetic flux density distribution in a rectangular YBCO
single crystal at T = 30 K and µ0 Hex = 171 mT (ZFC). The single crystal has a thickness of
20 µm and a lateral size of 1000 × 700 µm2 . The calculated flux (top) and current distribution
(bottom) for the partly penetrated state in a thick sample are also shown. It is necessary to take the
screening current in the flux-free regions into account to obtain the observed flux pattern (Schuster
et al 1995a).
dependence of the supercurrent, which would be desirable for a study of three-dimensional flux
penetration in thick samples, is non-trivial (see also section 3.4). Magneto-optical imaging
of the flux distribution on single crystals with a ratio of thickness to width of d/W ≈ 0.01–
0.1 (d λ) has been performed by several groups (Moser et al 1989, Koblischka et al 1990,
1995, Dorosinskii et al 1992, Forkl 1993, Turchinskaya et al 1993, Vlasko-Vlasov et al 1993a,
Schuster et al 1995a, Wijngaarden et al 1996). The magnetic flux pattern in the x–y plane of
the thick samples (Hex z ) is very similar to the flux density distribution observed in thin
films.
As an example, figure 29 shows the magnetic flux distribution at the surface of a 20 µm
thick YBCO single crystal (lateral dimensions ≈1000 × 700 µm2 ). As in rectangular or
square-shaped thin films, the magnetic flux shows a cushion-like pattern. The d + -lines which
begin to form a ‘double Y’ are visible as dark regions extended to the sample corners. In
comparison to the flux patterns in rectangular thin films, the peaks of Bz (x, y) at the d − - and
d + -lines are less pronounced. In addition, the slope of Bz at the flux front is much smaller.
Both observations are due to the increased sample thickness which reduces the height of the
self-field peaks and can be expressed on a macroscopic level by a reduced demagnetization
factor. For a detailed calculation of the thickness dependence of the flux profiles in the fully
penetrated state, see Forkl and Kronmüller (1994).
The penetration field in a strip with arbitrary thickness which can be useful for a estimate
of the magnitude of jc is given by Forkl (1993)
2 jc d 2W
d
W
Hp =
.
(60)
arctan
+ ln 1 +
2π d
W
d
For d W and d W , equation (60) has the limits (3) and (40) respectively. Equation (60)
may be useful also for rectangular samples.
The result of figure 29 (see also Wijngaarden et al (1996) and Schuster et al (1996a))
shows that a screening current j jc is required in the flux-free regions (no observable Bz component), in order to obtain the observed magnetic flux pattern. Moreover, the presence of
the screening current in flux-free regions has been established in type-II superconductors in
the perpendicular limit for specimen thicknesses from d = 100 nm (Jooss et al 1996a) up to
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Ch Jooss et al
a)
b)
Figure 30. Magnetic field lines during flux penetration into a thick strip with d/W = 0.25. The
external field is applied perpendicular to the larger surface. Its magnitude is Hex /Hp = 0.2, 0.4,
0.8 and 1 (a). In (b) the external field is decreased from Hp to Hex /Hp = 0.8, 0.6, 0.2 to the
remanent state Hex = 0. The bold curve shows the position of the flux front (Brandt 1996a).
250 µm (Frankel 1979). From these results the question arises of whether the screening current
is of reversible nature and represents the Meissner current of the flux-free central area of the
superconductor (Schuster et al 1994a, Glatzer et al 1992, Theuss et al 1992)—or represents
an irreversible critical current flowing in a limited (lens-shaped) area of the cross section due
to the three-dimensional flux penetration in samples of finite thickness (Prigozhin 1996, 1997,
Brandt 1996a).
Frankel (1979) pointed out that due to the observation of shielding currents while
decreasing Bex for a disc in a fully penetrated state, the shielding current cannot be attributed
to reversible magnetization properties. The presence of shielding currents during flux exit
from a fully penetrated state in a YBCO disc with d = 290 nm < λc was demonstrated by
Jooss et al (1998a); this also proves its irreversible nature. This result shows that, in thinfilm superconductors also, the shielding current cannot be attributed to reversible properties.
However, as will be analysed in section 5 and especially in section 5.5, Meissner currents
occurring at the phase boundary between the Shubnikov and Meissner phase in a type-II
superconductor with pinning are intrinsically irreversible. We will show in section 5 that the
shielding current for thin films is in fact a Meissner current, flowing in the entire cross section
of thin-film samples with d < λ.
The situation for thicker samples λ d W (still in the perpendicular limit), however,
is different. Theoretical results for the flux penetration in samples with finite thickness show
that an apparent shielding current may have its origin in the three-dimensional flux penetration,
where gradients of the in-plane components of B along z create critical currents which are
flowing in a surface region of the samples. In addition to some analytic calculations for
ellipsoids in the critical state (Krasnov et al 1991, Bhagwat and Chaddah 1992, 1994), there
have been analytical calculations of the field lines in a thick stripe (Forkl and Kronmüller
1994). Conner and Malozemoff (1991) considered the distribution of a field dependent jc in
a thick disc. Brandt (1996a) used the non-linear conductor equations given in section 4.3 to
calculate the flux penetration and current distribution for the two-dimensional problem of a
disc or an infinitely extended strip with finite thickness. In these simulations Hc1 = 0 and thus
Current distributions in high-Tc superconductors
705
vanishing Meissner currents have been assumed. An overview for various shapes and different
field geometries is given in Brandt (1998a); see also Prigozhin (1996), Brandt (1998b, 1998c).
An extension to finite Hc1 is given in Brandt (2001).
Figure 30(a) shows the magnetic field lines during penetration of perpendicular flux into a
thick strip with a thickness-to-width ratio of d/W = 0.25 for different applied fields. The flux
front (bold line) simultaneously indicates the current front, where the supercurrent is jumping
from j = 0 (flux-free region) to its critical value j = jc . For decreasing Hex (figure 30(b)), the
bold line indicates the sharp change of the supercurrent from +jc to −jc during the penetration
of the inverse flux.
Using equation (29) to calculate the thickness-averaged sheet current density from the
z-dependent j -distribution inside the specimen in figure 30, one recovers again the spatial
distribution of the shielding current. This was explicitly proven for an infinitely extended strip
with not-too-large thickness: to very good approximation one obtains the shielding current
distribution equation (37) which was calculated analytically for a thin strip with d → 0 (Brandt
1996a). In regions where j = jc in the entire sample thickness, the sheet current density has
the value J = jc d, whereas in regions where the magnetic flux has penetrated only a surface
area of the superconductor, one has J jc d. Note that in this surface regions jc is dominated
by ∂Bx /∂z or ∂By /∂z, respectively. Simultaneously, there is already a non-vanishing normal
component Bz , which is however reduced by a factor of w/d ≈ 102 –103 compared to the Bz values at the flux front. When considering samples with finite Hc1 , a non-vanishing Meissner
current would cause a modification of the lens-shaped flux penetration leading to a discontinuous change from finite Bz at the flux front to Bz = 0 inside the superconductor (see section 5.2).
In summary, for thick samples the shielding current is explained theoretically by the threedimensional ‘lens-shaped’ flux penetration in the critical state model of Brandt (1996a) which
is only slightly modified by introducing finite Hc1 or London penetration depths (Brandt 2001).
However, in thin films the Meissner current extends over the full width of the Meissner area,
flows in the entire cross section of the sample and cannot be distinguished from currents j < jc
which are resulting from a lens-shaped three-dimensional flux penetration. For further clarity,
especially for intermediate thicknesses, a systematic quantitative experimental study of the flux
penetration and current distribution as a function of the sample thickness would be strongly
desirable.
5. Meissner currents and surface barriers
Besides the irreversible magnetization and the corresponding magnetic flux and current
distribution in the non-equilibrium state which are due to vortex pinning, magneto-optics
has been used to investigate reversible properties of HTS, e.g. the Meissner expulsion, the
modification of the irreversible currents due to Meissner currents and, in addition, currents
being created by surface barriers. Since Hc1 is very small in HTS and one usually has flat
samples with strong demagnetization effects, the existence regime of a pure Meissner phase
in the (Hex , T ) phase diagram is very narrow. However, reversible properties can become
important also in the Shubnikov phase: it is shown in section 5.2 that the Meissner current,
which represents the response of HTS to Hex in the pure thermodynamic equilibrium state
will have irreversible characteristics when flux pinning is present in a superconductor. In
particular, flux pinning enables the non-equilibrium coexistence of two thermodynamic phases
in the partly penetrated state: a disordered Shubnikov phase and the Meissner phase, which
are separated under certain conditions by a Meissner current string at the phase boundary
(flux front) or at the Meissner hole which is formed by closed vortex loops. Also the currents
which are created by surface barriers (section 5.4) may exhibit irreversible characteristics,
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Ch Jooss et al
Figure 31. The flux pattern observed in a part of a Bi2 Sr2 Ca1 Cu2 O8 crystal after field cooling
in a perpendicular field of 5.6 mT to 5 K (a). The flux profile measured along the marked arrow
(x-axis) is plotted in (b). Negative x corresponds to the sample exterior (Indenbom et al 1994a).
since the barrier behaves differently for the entry and exit of vortices in the superconductor.
The Meissner effect in thin films will be treated in section 5.5. A short summary on flux
instabilities due to overcritical Meissner current is given in section 5.3.
5.1. Meissner expulsion
The Meissner expulsion of the magnetic flux in HTS samples during field cooling was
visualized by several groups (Vlasko-Vlasov et al 1992, 1993a, Dorosinskii et al 1993a,
Indenbom et al 1993c, 1994a). Figure 31 shows the magnetic flux distribution of a part of
a high-quality Bi2 Sr2 Ca1 Cu2 O8 single crystal with approximately rectangular cross section
after field cooling in a perpendicular field of 5.6 mT at 5 K. The largest Meissner expulsion is
visible at the sample borders as beltlike dark contrasts, revealing that the Meissner current is
mainly concentrated at the sample’s borders. In these regions, the driving force of the Meissner
current exceeds the pinning force and the flux is expelled from the sample. In the central part
of the crystal the flux remains trapped and the flux density is only slightly reduced with respect
to the external field. The observed inhomogeneous beltlike flux structure with a width of
≈100 µm λab is due to the large demagnetization effects.
Figures 32(a) and (b) show the same Bi2 Sr2 Ca1 Cu2 O8 single crystal after ZFC and
applying an external field of 5.6 mT for two different temperatures. During the increase
of the temperature, magnetic flux appears in the crystal centre and moves the Meissner phase
to its perimeter. In this example, vortices penetrate the Meissner phase along defect regions
near the top edge of the sample. However, such penetration can even take place in an ideal
crystal, where the Meissner screening current causes a driving force on the vortices which
exceeds the pinning force. For increasing temperature the same beltlike structure is created
in the ZFC experiments as in the field-cooled experiments—visible in figure 31. Since for
increasing temperature (above the irreversibility line of Bi2 Sr2 Ca1 Cu2 O8 ) the pinning force
on vortices completely vanishes, this proves that the beltlike structure is not related to flux
pinning.
The spatial distribution of the Meissner sheet current along a line crossing the centre of a
thin rectangular crystal or a thin strip is approximately given by
y
JM (y) ≈ Hc1
.
(61)
((W/2)2 − y 2 )1/2
Current distributions in high-Tc superconductors
707
a)
c)
b)
Figure 32. Flux penetration in the Bi2 Sr2 Ca1 Cu2 O8 crystal of figure 31 at µ0 Hex = 5.6 mT after
ZFC at (a) 5 K and (b) 49 K. (c) The temperature-dependent change of the flux profiles measured
along the same line as in figure 31 (Indenbom et al 1994a).
For the Meissner current in strips of finite thickness and solutions avoiding the singularity at
the strip edge, see Brandt (2000a). Indenbom et al (1994a) measured this distribution at the
86 µm thick sample visible in figures 31, 32 by the application of a special magneto-optical
technique, where the in-plane component By of the self-field is determined. This technique
is based on the condition that the domain boundary structure of the iron garnet indicator with
in-plane anisotropy is always at the position By = 0. By probing the position of the domain
boundary when it is moved by the application of an additional in-plane field, one is able to
measure the By (y) distribution reflecting directly the jx (y) component.
In particular, in field-cooled experiments, the value of the Meissner expulsion depends on
the pinning strength of the defects in the crystal lattice (Indenbom et al 1994a). This is due to
the trapping of flux lines which nucleate at material defects.
5.2. Modification of the critical state
The modification of the critical state by a finite Hc1 and the presence of Meissner currents has
been observed several times by means of magneto-optical techniques (Indenbom et al 1993a,
1993c, 1995, Dorosinskii et al 1994, Vlasko-Vlasov et al 1997, 1998). Figure 33 shows
the modified Bean model which has been proposed by Indenbom et al (1995) for a type-II
superconductor with pinning and finite Hc1 . For increasing external field Hex > Hc1 the
reversible Meissner expulsion splits up into two different jumps in the flux density distribution
which are related by Bean’s critical gradient. Both jumps are due to current strings which
have been recently reproduced in theoretical simulations (Brandt 1999a); for finite London
penetration depth see also (Brandt 2001). The first step Bs = meq Hc1 (meq denoting
the reversible magnetization normalized to Hc1 which can vary from 0 to 1) remains at the
sample edge. Bs is due to a current string of jM which is reduced for increasing B by
vortex eddy currents jv flowing in the opposite direction. This results in a decrease of
meq (H ) ∝ (jM (H ) − jv (H )) for increasing external field Hex > Hc1 . The step at the
surface Bs is therefore generated by the current string of magnitude jM − jv and decreases
708
a)
Ch Jooss et al
b)
µ-10 Bz
c)
Bz
Hex >H p
µ 0 H ex
jv
Bulk
jM
Hex>Hc1
meq H c1
y
<j v >
<j v >
j
0 -a
a
Edge
jc
Vortex
Bz
Flux front
0
µ 0 H ex
Thin film
y
Figure 33. (a) The modified Bean model for a superconductor in parallel geometry with finite Hc1
according to a model of Indenbom et al (1995). Flux density profiles and a current density profile are
shown after ZFC in increasing external field. (b) A sketch of the arrangement of the Meissner current
string jM , the vortices with microscopic eddy current density jv and the mesoscopic current density
jv at the surface and the flux front interface for parallel geometry. (c) The resulting averaged flux
density profile Bz (y). For comparison the flux profile for a thin film is depicted schematically. For
thin films Bs at the sample’s edge is hidden due to strong demagnetization peaks and the current
string at the flux front is developed to a macroscopically extended Meissner current distribution in
the flux-free area of the film (see section 5.5).
with increasing vortex density. Evidence for this surface step was found by Dorosinskii et al
(1994) by iron garnet-based magneto-optics. The temperature and field dependence of Bs
was utilized to distinguish carefully between contribution of the Meissner current and the
Bean–Livingston (BL) surface barrier.
The second step Bff occurs at the flux front ±P = ±(Hex − Hc1 )/jc and is caused by
surface-like currents which flow at the phase boundary between the Meissner and the vortex
phase. This step also occurs at the boundary between vortices with opposite orientation, e.g.
during flux reversal after application of a negative external field Hex < −Hc1 to the remanent
state of the sample. In this case the step at the flux front of the inverse flux penetration has the
double height 2 Bff compared to virgin flux penetration and the observation is more reliable
(see Indenbom et al (1995)). A step forward from a one-dimensional to a three-dimensional
consideration was made by Vlasko-Vlasov et al (1997). They showed in their experiments that
the remagnetization of bulk superconductors takes place by shrinking of closed induction loops,
forming a Meissner hole at positions where B < Bc1 which is centred at the annihilation front.
A three-dimensional model of this boundary, together with the experimental observations, is
presented in figure 34.
According to a model developed independently by Koppe (1966) and Campbell and Evetts
(1972), there will be a minimum radius RM of the vortex loops, determined by the balance of
the line tension force Ft = σ/RM and the pinning force Fp . In the isotropic case σ equals the
vortex line energy = Bc1 0 /µ0 . At small fields, Fp = jc 0 and RM = Bc1 /(µ0 jc ). The
experiments presented in figure 34 indicate a size of ≈7–15 µm in YBCO single crystals at
50 K. Loops of smaller radius will collapse, as for them Ft > Fp . As a result a cylindrical
flux-free region of diameter DM = 2RM is formed along the boundary and a Meissner current
density of jM ∝ Hc1 /λ is flowing in a layer with thickness λ along its surface. Such a current
between the Meissner and vortex phases is usually called the dB/dH effect (Campbell and
Evetts 1972).
Current distributions in high-Tc superconductors
709
b)
a)
c)
Figure 34. Top: (a) the scheme of magnetic field lines of a superconducting plate in a remanent
state; two flux-free cylinders where B = 0 are the Meissner holes; (b) detail of the Meissner hole
at a remagnetization front with diameter DM ; the critical currents Jc and the Meissner currents
IM are flowing in the same direction; (c) the shape of the Meissner hole in a square-shaped plate.
Bottom: magnetic flux patterns in a 40µm thick YBCO single crystal: (a) the remanent state after
field cooling in µ0 Hex = 150 mT and switching off the field at 20 K; the boundary between areas
magnetized up (⊕) and down () is close to the crystal edges; (b) application of −100 mT to the
state in (a); (c) the remanent state after field cooling in µ0 Hex = 150 mT and switching off the
field at 55 K; the Meissner hole is sharper and strongly bent; (d) application of −23 mT to the state
in (c) (Vlasko-Vlasov et al 1997, 1998).
The Meissner hole can be observed as a significant distortion of the normal flux density,
since the integral Meissner current iM is double the loss of the critical current inside the fluxfree cylinder (Vlasko-Vlasov et al 1997). Due to the locally enhanced current density in the
Meissner hole, a bending instability of the remagnetization front is occurring in a distinct
temperature range. This can be seen in figures 34(c), (d) as a meandering structure and is
one factor which leads to a phenomena called macroturbulence (see the next section). The
closed vortex loops which form the Meissner hole are already created at the end faces of the
plate before reaching the remanent state, if the external field is decreased in a magnetization
experiment. This can be seen in the model calculation of the field lines in figure 35(a). Endface observation of the Meissner holes and its movement towards the interior of the sample
are depicted in figures 35(b)–(e). Note that Meissner holes are also observed for long samples
in a parallel magnetic field (Vlasko-Vlasov et al 1998).
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Ch Jooss et al
Figure 35. The appearance of the Meissner hole on the end face of a magnetized plate in decreasing
external field (a) and observation of the Meissner hole at the end faces of a 110 µm thick YBCO
single crystal at 53 K. (b) µ0 Hex = 0 after application of 180 mT parallel to the c-axis. (c)
µ0 Hex = −17 mT. (d) and (e) show the (a, b) face images of the sample under the same conditions
as for (b) and (c), respectively. With increasing negative field, the Meissner hole moves inside the
sample and is smeared in the end-face observation (Vlasko-Vlasov et al 1997).
For the virgin magnetization after ZFC and applying an external field, no closed vortex
loops and consequently no Meissner holes can be formed. However, the sudden change of the
vortex density at the flux front from a finite value to zero gives rise to an inner interface, where
the eddy currents superimpose on a current string jv oriented along the flux front and decaying
on a length scale λ in the Meissner phase. The energy of this current string corresponds to a
difference between the chemical potentials of the vortices in the Shubnikov and the Meissner
phase. This suggests that the related flux step Bff is constant for an increasing penetration of
the flux front. This was confirmed by quantitative experiments on Nb single crystals (Albrecht
et al 2002 unpublished), where an Hex -independent current string was observed at the flux
front and is contradicts the statement Bff = (1 − meq )Hc1 made by Indenbom et al (1995).
In contrast to the Meissner hole in remagnetization experiments, the flux front current string
for virgin magnetization has not been clearly observed in HTS up to now. One observes rather
smooth Meissner current distributions in flux-free regions. Calculations of Brandt (1999a) for
this situation suggest that this current is decaying smoothly inside the lens-shaped flux-free
region of the Meissner phase in thick samples. In thin samples, d < λ, no current strings
are observed and the Meissner current extends over the full flux-free width of the films (see
section 5.5).
Current distributions in high-Tc superconductors
711
5.3. Macroturbulence and instabilities
Due to the overcritical current density of the Meissner holes at remagnetization fronts in bulk
HTS samples, this boundary can be subject to a bending instability as can be seen in figure 34.
The high current densities together with flux concentrations at the surface of the Meissner
hole assist the formation of macroscopic current loops, flux bundles and macrovortices which
move under the action of alternating external fields (Vlasko-Vlasov et al 1997), rotating fields
(Vlasko-Vlasov et al 1998) or due to thermal relaxation (Koblischka et al 1998c, Frello et al
1999) and temperature variation (Koblischka et al 1999b, 1999c) over macroscopic distances.
These macroturbulent features which are formed at the interface region of vortices with different
orientations have to be well distinguished from other instabilities due to local heating caused by
non-linear vortex dynamics. Local heating and other mechanisms can also lead to instabilities
(see e.g. Paltiel et al (2000)) and apparently similar features, such as kinetic flux front
roughening (Surdeanu et al 1999), magnetic avalanches (Nowak et al 1997, Vlasko-Vlasov et al
2000) and dynamic dendritic structures (Leiderer et al 1993, Duran et al 1995, Bolz et al 2000,
Johansen et al 2001). To our knowledge, flux jumps and dendrite branching were first observed
by Wertheimer and Gilchrist (1967). For a recent theoretical study see Aranson et al (2001).
5.4. Surface barriers
The critical magnetic field for first flux entry into a superconductor can deviate significantly
from Hc1 due to surface barriers. One of the most general is the BL surface barrier
(Bean and Livingston 1964) which enhances the first penetration field to a value of Hp,BL
(Hc1 H Hp,BL Hc2 ). Due to this microscopic surface barrier, the vortices are repulsed
from the sample surface for virgin magnetization during field increasing. This results in a
delayed reduction of the Meissner current string jM at the surface and thus in an increased
equilibrium magnetization for H > Hc1 . If the Shubnikov phase is established in a sample,
for decreasing fields the BL barrier causes an attractive force on the vortices near the surface
and assists the flux exit. In this case the reduction of the Meissner current string jM with
decreasing field is more pronounced and thus Bs is reduced. Dorosinskii et al (1994) found
a strong asymmetry of Bs for increasing and decreasing external fields in a YBa2 Cu3 O7 single
crystal at low temperatures T < 35 K. The effect of the BL barrier at higher temperatures
is strongly reduced by the increasing flux creep rate, weak places and inhomogeneities at the
sample surface. In HTS with strong pinning, where the attractive or repulsive surface forces
are small compared to the pinning forces, the BL surface barrier was not observed by means
of magneto-optical measurements.
In addition to the microscopic BL barrier for straight vortices penetrating a parallel surface,
there is a similar macroscopic geometric barrier for flux penetration at the edges, e.g. in
samples with rectangular cross section (Zeldov et al 1994c). For a theoretical description,
see Brandt (1999b, 2000b). For rectangular cross section, the entrance field is given by
Ben ≈ Bc1 tanh(0.36d/W )1/2 (Brandt 1999a). The behaviour of samples with geometric
barriers is partly similar to that of samples with artificially enhanced pinning near the edge
(see section 6.5). Experimental evidence for a geometric barrier at temperatures near Tc for
Bi-2212 single crystals is given by Zeldov et al (1994c). Evidence that a geometrical barrier
reduces the losses of alternating transport currents in HTS films is reported by Kerchner et al
(1999). An enhancement of the surface barrier by soft magnetic environments in YBCO films
is reported by Jarzina et al (2002).
712
Ch Jooss et al
Figure 36. (a) An optical image of the YBCO sample (d = 250 nm) showing the positions of the
laser patterned slits. (b) The magneto-optical image of the flux patterns at µ0 Hex = 10 mT after
ZFC. The length of the horizontal flux pattern at the slit is ≈0.5 mm. (c) A surface plot of the flux
density distribution of the twin lobe of the horizontal slit and (d) current streamlines around the
slit as obtained by inversion of the Biot–Savart law (Baziljevich et al 1996b).
5.5. Reversible properties in thin films
For thin superconducting samples (d λ) in a perpendicular magnetic field or with a transport
current, there are two serious modifications of the Meissner state compared to the case for
bulk samples. (i) An observation of full, reversible flux expulsion for the entire sample,
corresponding to a pure Meissner phase, is only possible for Hex Hc1 . (ii) Since the
thickness is too small, no Meissner current strings or Meissner holes are developed and the
Meissner currents redistribute over the full sample width.
Let us first consider the field of first flux penetration in a thin film: due to the large
demagnetization effects, the self-field of the Meissner current distribution (see equation (61))
in thin films has a large maximum near the film edges (see e.g. figure 36). Consequently,
even for Hex < Hc1 , the magnetic self-field at the edges exceeds Bc1 for d λ and flux lines
start to penetrate the film. Actually, to our knowledge, in ZFC experiments no observation of
a fully reversible Meissner expulsion is reported for epitaxially grown thin films of HTS in
the thickness range up to 500 nm. Only for a 2 µm thick epitaxial film has a full Meissner
expulsion been observed (Jooss 2000). Jarzina et al (2002) achieved suppression of flux entry
and stabilized the Meissner state in a 1.2 µm thick epitaxial YBa2 Cu3 O7 film by soft magnetic
screenings. In field-cooled experiments, the Meissner expulsion is suppressed due to flux
trapping at strong pinning sites in the films. In ZFC experiments on thin films, the strong
pinning assists the observation of the coexistence of Meissner and vortex phases in a relatively
large field range below the full penetration field Hp (see e.g. figures 14, 21 and 22).
In thin films the two steps Bs and Bff due to the current strings described in section 5.2
for bulk samples and also the Meissner holes at the remagnetization front are not observed.
Due to the strong curvature of the magnetic field lines and because the diameter of a current
Current distributions in high-Tc superconductors
713
string ≈2λ would be larger than d, the Meissner current density has to redistribute and extends
over the entire film cross section which is in the Meissner state. This can be seen nicely
in figure 36, where the Meissner current flows in the entire dark (flux-free) region (see also
figures 14, 21 and 22). Besides the model-independent determination of the Meissner current
density distribution in these figures, a direct proof for the macroscopic extension of the Meissner
current in thin films is given by the artificial distortion of the Meissner current shown in
figure 36 (Baziljevich et al 1996b). Due to its suppression at the slits, the Meissner current has
to redistribute and creates peaks of the magnetic flux density at the slit regions which penetrate
into the superconductor. Note that the magnetic field lines of these flux lobes are closed and
consequently the corresponding vortices have been nucleated at the slits and not at the sample’s
edges.
Interestingly, there still remains a steplike feature in Bz at the flux front which resembles
Bff . In contrast to the case for Bff for thick samples, in thin films the large slope of the
normal component Bz at the flux front (see figures 14 and 21) is not related to a current string.
It is simply related to curvature of the field lines and thus to the extended irreversible Meissner
current in areas where Bz is vanishing. In the limit of vanishing thickness, the slope diverges:
∂Bz /∂y|y=Q → ∞.
Comparing bulk with thin-film samples, the surface step in the normal component of the
flux density Bs is also not observed in thin films and is possibly hidden behind the stray-field
peak. In thin-film remagnetization experiments, a shielding current which corresponds to the
reversed flux and critical current density is developing over the entire film (Jooss et al 1998a)
(see e.g. figure 39), and this is in contrast to the concentration of the Meissner currents in
the Meissner hole as observed in bulk samples. This was already suggested by Brandt and
Indenbom (1993) for strips. The Meissner screening current of the remagnetization front has
to be superimposed on the critical current of the region of virgin magnetization, thus leading
to a non-local variation of the current distribution throughout the entire specimen.
6. Anisotropic and inhomogeneous currents
In section 4, the current distributions for different sample geometries have been considered
only for the case where the pinning strength Fp is uniform in the HTS samples. This means that
jc in the flux-filled parts is constant (if the electric field dependence jc (E), the magnetic field
dependence jc (B) and the Meissner currents are disregarded). In the following, the condition
jc = constant in the flux-filled regions is abandoned and an analysis of the current domains
and d-line patterns created by anisotropic and spatially inhomogeneous jc is performed. This
is a necessary step for a full understanding the current flow in superconductors with complex
microstructures such as tapes and wires (section 7) and, in addition, for a study of vortex
pinning mechanisms in HTS, as will be seen in section 8.
In this section, we first consider the case of an anisotropic current density, where the
magnitude of jc depends on the direction of the current flow (section 6.1). Afterwards, the
critical state with a field-dependent critical current jc (B) is considered, where the magnetic field
gradients within the superconductor lead to a continuous spatial variation of jc (section 6.2).
Next, samples with two areas of different vortex phases and with different vortex pinning
strengths are considered in sections 6.3 and 6.4, respectively. The resulting current distribution
for j n is discussed in section 6.4 (n denotes the normal vector of the domain boundary
between areas of different jc ). For j ⊥ n two different cases have to be distinguished, since the
developing current distribution dramatically changes if magnetic flux first penetrates the current
domain with higher j2 and this is followed by a penetration into the low-j1 domain, or vice
versa. Section 6.5 describes the first case where j2 is located at the border and j1 at the centre
714
Ch Jooss et al
of the sample. Finally, section 6.6 considers the second case. Since variations of the pinning
strength are obtained by different methods, such as partial sample thinning (section 6.4), heavyion irradiation of a part of a sample (section 6.5) and substrate modifications (section 6.6),
each method is briefly summarized in the corresponding sections.
6.1. Anisotropic critical currents
Anisotropic critical currents may have different microscopic causes in HTS. First of all, the
layered crystal structure, where the superconducting charge carriers are mainly concentrated
within the CuO2 planes, gives rise to an intrinsic anisotropy of jc,ab flowing parallel to the
a–b planes and jc,c flowing parallel to the c-axis (see e.g. the review of Senoussi (1992)).
This anisotropy is mainly caused by the different superconducting couplings in the two
directions but also is influenced by different pinning strength of interlayer (Josephson) vortices
compared to that for Abrikosov or pancake-like vortices parallel to c-direction. It may be
influenced by demagnetization effects in thin-film geometries (Theuss et al 1994). Several
authors have already measured the anisotropy ratio jc,ab /jc,c by means of magneto-optical
techniques and obtained very different values: with melt-textured samples Schuster et al
(1993) found a value of 25, whereas Gotoh and Koshizuka (1991) found 3 and Jooss et al
(2000c) a value of 2, indicating that the ‘intrinsic’ anisotropy is strongly modified by the
microstructure (see also section 7.3). For YBa2 Cu3 O7 single crystals, Polyanskii et al (1990a)
found a value of 7 and a similar value is observed by Cuche et al (1996), however emphasizing
that planar defects parallel to the a–b plane are also present in single crystals.
Anisotropic jc s are also observed due to an anisotropic microstructure: planar defects,
such as antiphase (Haage et al 1997, Jooss et al 1999, 2000a) or twin boundaries (Zandbergen
et al 1990b, Wijngaarden et al 1997) exhibit strong anisotropic pinning forces (see section 8.2).
A local anisotropy of jc was observed by magneto-optical measurements in parts of YBCO
films grown on substrates with ion-irradiated line patterns (Leonhardt et al 2000b). In addition,
anisotropic depinning mechanisms have been observed at columnar defects which are tilted
with respect to a superconductor surface (Schuster et al 1995c) (see section 8.1). Furthermore,
anisotropic jc s may be induced by strong longitudinal magnetic fields due to the vortex–vortex
interaction (Indenbom et al 1994b, Schuster et al 1995d).
Anisotropic current distributions can be studied nicely in square-shaped samples in
perpendicular external fields, since the pattern of the d + -lines allows a direct determination
of the anisotropy ratio Aj = j2 /j1 , where j2 indicates the high jc -value and j1 the lower one
(Forkl and Kronmüller 1995, Schuster et al 1997). Figure 37(a) shows a schematic diagram
of the current domain structure together with the d + -lines separating the j2 - and j1 -domains.
Instead of an ‘X-like pattern’ as observed in a square sample with isotropic jc , the d + -lines
form a ‘double Y-like pattern’ (Gyorgy et al 1989, Sauerzopf et al 1990). From div j = 0 it
follows that
bj1 = aj2 ,
(62)
which determines the size a of the current domain with higher jc , whereas b is fixed at W/2.
The anisotropy ratio of the currents can be easily determined with high precision from
j2
= cot α,
(63)
Aj =
j1
where α denotes the angle of the d + -lines with respect to the edge j2 .
Figure 37(b) depicts an experimental result for a YBCO film with j2 = jy = (2.0–
2.6) × 1011 A m−2 and j1 = jx = (0.8–1.2) × 1011 A m−2 , corresponding to a mean
anisotropy ratio of Aj = 2.3. The anisotropy in this film is due to parallel-oriented antiphase
Current distributions in high-Tc superconductors
a)
715
b)
j1
j2
j2
j1
Figure 37. (a) A sketch of the current distribution and discontinuity lines for an anisotropic critical
current density with a component of high magnitude, j2 , and a component of low magnitude, j1 .
(b) The flux density distribution in a square-shaped, 320 nm thick YBCO film with anisotropic
critical current density at Bex = 48 mT and T = 5 K. Full lines indicate the current path and,
in addition, the current density profiles are plotted as the dashed lines. Three dark spots indicate
distortions in the MOL.
boundaries (y) being created by film growth on vicinal substrates (Haage et al 1996, 1997).
These parallel-oriented planar defects with a mean distance of ≈6 nm lead to the observed
anisotropic pinning force. Further examples with anisotropic current densities are shown in
figures 54 (intrinsic anisotropy) and 79 in this paper. A theoretical solution for the flux and
current distribution in thin strips with anisotropic or field-dependent pinning is given by Mikitik
and Brandt (2000a) and for arbitrary shape by Mikitik and Brandt (2000b).
6.2. Field-dependent critical currents
A deviation from the Bean assumption jc = constant and thus a spatially inhomogeneous
critical current may also appear due to a local magnetic field dependence of jc (B(x, y)) due to
flux density gradients within superconducting samples (McDonald and Clem 1996). Such an
effect can be observed in the magnetic field range available to magneto-optics only in samples
with a strong low-field dependence of jc (B) or/and large flux gradients. This is demonstrated
in the following by a magneto-optical study of the flux and current distribution in a YBCO
thin disc with a diameter of 2 mm and a film thickness of 290 nm (Jooss et al 1998a).
In order to rule out an apparent spatial dependence of jc due to calibration errors (see
section 3.3), the measurements of the Bz -distribution are performed with EuSe as the MOL.
The external field is applied perpendicular to the film plane and is increased from the ZFC
state (Hex = 0) to a maximum field Hex,max . Afterwards it is successively decreased until
one approaches the remanent state (Hex = 0). Figure 38 shows the magnetic flux distribution
and the current flow lines together with a current density profile of the thin disc. We use
two-dimensional polar coordinates r = (r, φ, z = 0) in this section.
With increasing external field, the flux penetrates radially, starting at the disc’s border.
Figure 39 shows the profiles of the magnetic induction Bz (r) and of the angular current density
jφ (r) for different applied external fields, increasing from the ZFC state. Note that the current
716
Ch Jooss et al
a)
b)
Figure 38. The magnetic flux density distribution together with the current flow lines and a profile
of the jφ (r) distribution of a YBCO disc at 4.2 K. (a) µ0 Hex = 48.8 mT after ZFC and (b)
µ0 Hex = 47.2 mT after application of a maximum field of 176 mT. MOL: EuSe.
a)
b)
Figure 39. (a) Profiles of the Bz (r) and jφ (r) distributions averaged over an angle of 70◦ . The
external field µ0 Hex is successively enhanced from the ZFC state to 16, 48.8, 112 mT and finally
to 176 mT, which is the maximum applied external field. The radial profiles are plotted in a
mirror-inverted fashion for r < 0. (b) Averaged profiles of Bz (r) and jφ (r) for decreasing external
fields demonstrating the flux reversal. The external field µ0 Hex is successively reduced from
µ0 Hex,max = 176 to 112 mT and 47.2 mT.
distribution of the disc is deviating from a pure one-dimensional jφ (r) because the presence of
macroscopic defects creates jr (r, φ) components of the current density. The current density
distribution j = (jr (r, φ), jφ (r, φ), 0) with the radial and the angular components jr and jφ
is obtained using the two-dimensional inversion of the Biot–Savart law given in section 3.4.
The radial profiles Bz (r) and jφ (r) are averaged over an angle of 70◦ (in the left area of the
disc in figure 38) to suppress noise and deviations due to large precipitates (≈1–10 µm) in
the sample. Following the Bean model (Bean 1964) or a modified Bean model for thin discs
(Mikheenko and Kuzlovlev 1993, Zhu et al 1993), one expects a constant critical current
Current distributions in high-Tc superconductors
717
Table 3. Parameters of the Kim model as obtained by fitting equation (64) to experimentally
obtained local and global field dependences of jc .
Measurement
jc,0 (A m−2 )
B0 (T)
jc (Bz,loc )
M(Hex )
2.98 × 1011
0.158
1.07
1.73 × 1011
density in the flux-filled part of the disc, forming a plateau jc = constant. However, as
shown in figure 39, the jc -plateau is not spatially constant. The critical current density is
decreased in areas of larger local flux density which are located at the disc borders for increasing
external fields.
Figure 39(b) shows the same flux and current density profiles as in figure 39(a) for
decreasing external fields. When the external field is reduced from its maximum µ0 Hex,max =
176 mT, the magnetic flux density exits through the disc borders. Consequently, a zone
where the magnetic flux density is dragged off penetrates into the disc. Due to the inverse
flux density gradients created, a zone with oppositely directed current density −jφ is then
penetrating towards the disc centre.
Again, −jφ (r) in figure 39(b) is not spatially constant. Generally, 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 and jc (B) is lowered there. For decreasing external fields the
situation is just the opposite. The local magnetic field shows a minimum at the disc border and
jc (B) is enhanced. These findings correspond to a universal dependence of the critical current
density jc (B) on the local magnetic flux density which is depicted in figure 40(a). The data
were obtained by combining Bz (r) and jφ (r) in figure 39 by eliminating the r-coordinate. All
jc (Bz ) for different Hex give a universal field dependence of jc ; in particular jc is suppressed
monotonically with higher local flux densities.
The observed jc -distribution in thin films for a field-dependent jc (B) was also predicted by
computer simulations of the flux and current distribution in thin films with a field-dependent
critical current density (McDonald and Clem 1996). A model often used to describe field
dependencies of the critical current is the Kim model (Kim et al 1962). It assumes the following
field dependence of the critical current density:
jc (B) = jc,0
1
,
1 + B/B0
(64)
where B0 is a constant field that characterizes the degree of field dependence and jc,0 is the
current density at zero field. Fitting the Kim model to the local field dependence in figure 40(a),
one obtains B0 = 0.158 T and jc,0 = 2.98 × 1011 A m−2 .
Usually, the magnetic field dependence of jc in superconductors is measured as a function
of the applied external field Hex by means of magnetometers or transport currents. If Hex
is much higher than the self-field being generated by the currents in the sample, one may
neglect the field gradients within the sample and the magnetic field dependence of jc is given
by the dependence of the global average current density j¯c on Hex to a good approximation.
However, this does not hold for small magnetic fields: comparing the local jc (Bz ) with the field
dependence of the global average current density j¯c as determined by SQUID measurement, one
finds significant differences (see figures 40(a) and (b)). The global j¯c (Hex ) is calculated from
the magnetization hysteresis M using (Bean 1964, Fietz and Webb 1969)
M(Bex )
j¯c (Bex ) =
kR
(65)
718
Ch Jooss et al
a)
b)
Figure 40. The magnetic field dependence of jc and the magnetization of the YBCO disc in figure
38: (a) the dependence of the local jc on the local normal component of the flux density Bz,loc for
different increasing (↑) and decreasing (↓) external fields. The smooth solid line represents a fit of
the Kim model to the jc (Bz,loc ) data (see table 3). The dashed line corresponds to j¯(Bex ) obtained
from (b), where the hysteresis of the magnetic moment µ0 M of the disc is presented, as measured by
a SQUID magnetometer. In addition, the magneto-optically determined µ0 M is depicted, which is
calculated from the current distribution. The maximum applied fields are ±200 mT for the SQUID
and 176 mT for the magneto-optical measurement. Note the nice agreement of the curves within
experimental error.
with disc radius R and a geometry factor k = 2/3 for a disc. In figure 40(a) jc (Hex ) is plotted
as a dashed line. It is clearly visible that the global current density is significantly smaller for
µ0 Hex < 100 mT than the local current density obtained by magneto-optics. Note that the
magnetization curve calculated from the local current density distribution using d2 r r × j (r )
is quantitatively in good agreement with the magnetization obtained by the SQUID. In table 1
these values of the fit parameters B0 and jc,0 are listed for the local and the global measurements.
Both, jc,0 and B0 obtained by magnetization measurement are significantly smaller compared
to the values obtained by local measurements.
The reason for the underestimation of j¯c by the SQUID for small Hex is that one assumes
a magnetization current density jc flowing uniformly in the entire sample. For the current
distribution during flux reversal this assumption is, however, strongly violated. Furthermore,
a different field dependence of jc appears due to the different values of the local flux density
B(x, y) and µ0 Hex .
6.3. Vortex phase transitions
In addition to the local magnetic field dependence of jc , a spatial variation of jc is observed
due to the coexistence of different vortex phases at different local flux densities (Yeshurun et al
1999, Giller et al 2000a, 2001a). For Bi-2212 single crystals, magneto-optical experiments
imply the coexistence of high- and low-current phases corresponding to disordered and quasiordered vortex phases, respectively (Giller et al 2000a); for theory, see Giller et al (2001b).
The time evolution of the vortex phases is shown by Giller et al (2000b); see also Giller et al
(2000c). The liquid–solid transition in the vortex lattice was visualized by ultrahigh-sensitivity
magneto-optical imaging by Soibel et al (2000).
6.4. Domain boundaries with perpendicular currents
In superconductors with anisotropic current density, the change of the value of j is intrinsically
related to a change in the current orientation leading to current domain boundaries where |j |
and ĵ are discontinuously varied. In contrast, for continuously varying |j |, e.g. for a local field
Current distributions in high-Tc superconductors
719
dependence of jc (B), or continuously varying transition temperatures (Gaevski et al 1998),
no splitting of the current pattern into domains of different magnitudes can occur. We will
consider now the case where the magnitude of the current density is sharply varying in an area,
where the boundary condition for the sample requires a uniform current flow parallel to the
sample edge. In general, two different cases may be distinguished:
(1) A boundary between two areas of different j1 and j2 which is oriented perpendicular to
the current (normal vector n parallel to j ). This boundary is related to complex Bz (x, y)
and j (x, y) patterns and leads to an enhanced dissipation at the boundary region. It will
be considered in this section.
(2) A boundary between two areas of different j1 and j2 which is oriented parallel to the
current (normal vector n perpendicular to j ); see sections 6.5 and 6.6.
In this section we focus now on case (1). Investigations of the flux distribution in
type-II superconductors show a preferential flux penetration along boundaries separating
superconducting regions with different critical currents (Schuster et al 1992b, 1993a). These
special features at the boundary lead to complex deviations from the flux penetration into
homogeneous specimens of arbitrary shape. For a simple experiment with a definite boundary
with normal vector n j , a rectangular YBCO single crystal with side ratio b/a = 3 and
thickness d = 40 µm was used (Schuster et al 1995a); see figure 41. The sample was
partitioned into three equal parts with square shape. In order to induce an inhomogeneity in
the critical sheet current, both outer parts where thinned down to about (2/3)d. The boundary
between the different j2 and j1 is located at a straight sample edge, where the current would
flow parallel to the sample edge in the limit j2 = j1 .
Figures 41(a)–(c) show the calculated current and flux pattern and the experimentally
determined flux distribution for this crystal at T = 20 K for an external magnetic field of
µ0 Hex = 213 mT. The experimental image was obtained by an iron garnet indicator. The black
spots on the images are defects in the indicator film. In figure 41(b) the contour Bz = µ0 Hex
is emphasized as a bold line. The different sheet currents are indicated by different streamline
densities. In the region with J2 (=1.5J1 ), the central streamline loops do not intersect the
boundary, at which Jc jumps.
Since the streamlines bend not only at the sample’s corners but also in the central area near
the jump of Jc , one observes a complicated d-line structure in the critical state which is depicted
in figure 41(d). In addition to the d − -lines at the sample edges one has a d − -line running along
the boundaries at which the sheet current changes from J2 to J1 . In the central region, additional
d + -lines run along the lines which connect the bending points of the streamlines. The ratio
J2 /J1 can be determined from the angle α between the d − -line along the boundary and the
d + -line running into the central region via J2 /J1 = 1/ cos 2α = 1.5 (Schuster et al 1994a).
The other d + -lines within the sample are due to the rectangular geometry of the whole crystal.
For more details and the development of the flux pattern with different external fields, see
Schuster et al (1995a).
If one determined the ratio J2 /J1 from the different flux penetration depths (Brandt
et al 1993, Brandt and Indenbom 1993, 1994, Norris 1970) on either side of the boundaries
separating regions with different Jc , one would obtain a value of about 2, which is larger than the
thickness ratio of 1.5. This apparent contradiction can be resolved by considering the current
flow through such a boundary: the larger sheet current J2 coming from the central part bends in
order to satisfy the continuity equation div J = 0, such that the normal component of J through
the boundary equals the lower sheet current J1 . The resulting jump of the tangential component
of J along the boundary then produces a sharp positive ridge of the magnetic field Bz on top of
the smooth field which is generated by the entire current distribution and by the external coil.
720
Ch Jooss et al
a)
d)
b)
e)
c)
Figure 41. Calculated current (a) and flux patterns (b) (contour plot) and magneto-optically
determined flux distribution (c) in a rectangular YBCO single crystal with inhomogeneous sheet
current Jc (x, y) for µ0 Hex = 213 mT. The critical sheet current in the left and right parts is about
a factor 2/3 lower than in the central region. (d) Visualization of the enhanced flux flow or electric
field along the boundary where the critical current density jc or the critical sheet current Jc abruptly
change. The bold lines denote the current flow and the dashed lines the flow of the magnetic flux
lines. (e) A 3D plot of the electric field distribution E(x, y) of the inhomogeneous sample shown
in figure 41 when Hex is increased further after full penetration. The straight lines along the sample
edges mark E = 0 (Schuster et al 1995a).
By the same token a depression of Bz occurs at the two d + -lines adjacent to this boundary. For
finite electric fields, however, the tangential component of J is changing continuously (holding
∇×E = −Ḃ ) and the ridge in Bz is slightly blurred. Combined, these three negative–positive–
negative ridges of Bz mean a concentration of flux which is seen in figure 41(c) as a deeper
flux penetration along the boundary, particularly into the region of lower Jc .
Another interesting effect which can be derived from the d-line structure in the fully
penetrated critical state concerns the velocity of the penetrating flux, which diverges along
the boundary, where Jc changes abruptly. The region around the boundary where Jc jumps is
depicted, enlarged, in figure 41(d), in which a ratio of 2 between the sheet currents has been
chosen (1.5 in figure 41) in order to get a larger angle α = 30◦ for clarity. The streamlines are
drawn as bold lines. Our area of interest is the enclosed triangular region between the d + - and
d − -lines, where the current does not flow parallel to the sample’s edge. The flux lines move
perpendicular to the current flow towards the d + -lines but do not cross these lines as indicated
by the dashed arrows. This means that the flux lines can penetrate into the triangular region
only along the boundary and from there along lines perpendicular to the current streamlines
towards the right and upper d + -lines. All flux lines which flow into this triangle have to
pass through the point where the boundary hits the specimen edge, but fewer flux lines pass
through the inner points of the boundary. As a consequence, E(x, y) exhibits a maximum at
the boundary position.
Current distributions in high-Tc superconductors
721
A 3D plot of the electric field distribution E(x, y) = |E (x, y )| calculated by the method
of Brandt (1995b) already described in section 4.3 is depicted in figure 41(e) for the critical
state with further increasing Hex . The ridge of E along the boundary shows a shape like an S.
One finds that E is maximum at the d − -lines while at the d + -lines we have E = 0. This is in
agreement with the statement that flux lines can cross only the d − -lines and not the d + -lines.
For a model-independent determination of the current density distribution at a current
domain boundary with perpendicular currents, see also figures 85(a) and (b) in section 8.3.2.
To summarize, the enhanced flux penetration in inhomogeneous samples at the boundary
separating regions with different critical currents is well reproduced by the non-linear conductor
model of Brandt (1995b). The deeper flux penetration along this boundary can be explained by
the ridgelike maximum of the magnetic field induced by the sudden change of the tangential
component of the current at the boundary, together with the concentration of the electric field
along the boundary. An electric field concentration means a high flux flow rate and therefore
a high dissipation of energy. This can trigger thermal flux jumps, which should be avoided in
high-current applications of superconductors.
6.5. Domain boundaries with parallel currents: I
We consider now the case of an inhomogeneous superconductor with two different current
densities j1 and j2 , where the domain boundary between the current domains of different jc is
oriented parallel to the currents. First, we consider the case where the region of higher j2 is
located near the sample border and consequently magnetic flux penetration takes place into the
j2 -domain first, followed by the penetration into the j1 -domain which is located in the inner
area of the sample (Schuster et al 1994c, 1994d).
The spatial variation of jc was introduced in a Bi2 Sr2 CaCu2 O8+δ (Bi-2212) single crystal
which was prepared with a thickness of about 30 µm as described by Li et al (1994). The highcurrent domain was prepared by heavy-ion
√ irradiation of definite areas in the sample, creating
linear defects (LDs) of radius R ≈ 2ξ (ξ = coherence length) which can remarkably
enhance the critical current density jc of superconductors in the irradiated areas (Civale et al
1991, Konczykowski et al 1991, Hardy et al 1991a, Gerhäuser et al 1992, Leghissa et al 1992,
Schuster et al 1992b, 1992c) by factors of at least 20–50. The irradiation was performed
through a mask by 860 MeV Xe ions at GANIL (Caen, France). At T = 50 K the unirradiated
sample was probably above the depinning line (Schuster et al 1994c). During the irradiation,
the centre of the sample was covered by an absorber to expose just the outer regions of the
sample to the ion beam (see figure 42(a)). Later the absorber was removed by an organic
solvent. For a short description of pinning at columnar defects, see section 8.1.
A significant feature of flux penetration into superconductors in the perpendicular limit
in increasing perpendicular field is the appearance of the supercurrent immediately over the
whole sample surface (see section 4, e.g. figure 13). The value of this current is always highest
at the sample edges and, if the superconductor is homogeneous, the front of the saturation of
the current is always followed by flux penetration. But if the critical current density jc in the
edge zone is higher than in the centre, the screening current can reach the lower jc there before
flux starts to penetrate. What will happen in this case? Two scenarios are conceivable:
(a) Zero (or reduced) sheet current J in the central zone may be forced by the nucleation
of parallel vortices generated by J and oriented along the two flat surfaces in opposite
directions. If these vortex–antivortex pairs are not pinned they will move into the specimen
and form vortex loops which do not violate the conservation of total flux in this central
region; appropriate curvature of these vortices can then compensate for the Meissner
722
Ch Jooss et al
Figure 42. (a) The shape of the irradiated Bi-2212 single crystal. The absorber is visible as a bright
region at the sample centre. (b) The flux distribution at T = 50 K in a perpendicular magnetic field
of µ0 Hex = 85 mT and (c) µ0 Hex = 107 mT. The flux starts to penetrate the unirradiated part of
the sample at the narrowest point of the irradiated belt (the white arrow in the lower left corner)
and piles up in the sample centre as indicated by the other white arrow. (d) µ0 Hex = 128 mT. (e)
µ0 Hex = 149 mT. (f ) µ0 Hex = 171 mT. (g) µ0 Hex = 213 mT. (h) µ0 Hex = 277 mT. The black
lines indicate the sample edge.
screening current. The resulting situation J = 0 is equivalent to a superconducting ring.
The magnetic field lines then close around the current-carrying edge zone.
(b) Alternatively, the sheet current J may exceed the J1 -value in the central region.
Consequently, the central region is completely screened by the shielding current as long
the flux front remains in the J2 -domain. If the first vortices arrive at the d − -line which
separates the two current domains, they are driven into the sample centre by the Lorentz
force which is generated by the shielding current J > J1 . We will see that the experiments
are consistent only with this latter assumption of overcritical shielding currents in the J1 domain (Schuster et al 1994c).
Figure 42(a) shows the shape of the Bi-2212 single crystal and the location of the absorber,
the bright area in the sample centre. In the sequence of figures 42(b)–(h), flux distributions are
presented at temperature T = 50 K for different applied perpendicular fields. In figure 42(c)
the flux front reaches the unirradiated area at the narrowest point of the irradiated belt (the
Current distributions in high-Tc superconductors
a)
723
b)
Figure 43. (a) Flux density profiles taken from figures 42(b)–(h) along the line indicated by two
arrows in figure 42(h). (b) Profiles of the sheet current J (r) and perpendicular magnetic field
H (r) in a circular disc with radius a and thickness t a with irradiation-enhanced critical current
density j2 = j0 in the ring zone r0 r a and j1 = 0.05j0 in the unirradiated central zone
0 r < r0 . r0 = 0.6a. J and H are in units of J0 = j0 d. The current–voltage law was assumed
as E = (j/jc )19 Ec with jc = jc (r, |H |) chosen such that jc j0 in regions where |H | < 0.02
(Meissner state). Increasing Hex = 0.1, 0.2, 0.3, 0.4, 0.5, 0.62, 0.66, 0.72, 0.78, 0.84, 0.9. The
inset shows the corresponding magnetization curve.
white arrow in the lower left corner in figure 42(c); see also figure 42(a)). Now an interesting
phenomenon occurs: flux suddenly appears in the centre of the Meissner area. At the same
time, the motion of the flux front from the sample edges slows down and the magnetization is
dominated by the spread of flux from the centre. Since vortices cannot nucleate at the sample
centre, they have to cross the Meissner area being driven to the centre by screening currents
which are much higher (‘overcritical’) than the j1 in this unirradiated region; figures 42(d)–(f ).
This situation is analogous to the penetration of flux bundles over an edge barrier observed in
type-I superconductors (Hübener et al 1972) and recently in Bi-2212 single crystals (Indenbom
et al 1994a). In figures 42(g), (h) the unirradiated part of the sample is completely filled with
flux lines and the flux front has moved from the sample edge to the inner edge of the irradiated
belt. In the irradiated belt only the critical current flows and further increase of Hex does not
change the shape of the flux density profiles but shifts them rigidly at the same rate as Hex .
The appearance and growth of the magnetic flux in the sample centre is also seen in the
flux density profiles measured across the sample; figure 43(a). These profiles nicely agree
with the calculated field profiles of figure 43(b), which are calculated in the framework of the
non-linear conductor model of Brandt. Moreover, the ‘uphill motion’ of flux lines, predicted in
Brandt (1992b, 1992c), is clearly seen: during the growth of the central flux heap the arriving
flux lines move against the flux density gradient since the driving current in this geometry is
caused mainly by the curvature of the flux lines.
724
Ch Jooss et al
At this point we come back to the discussion of flux line motion within the framework of dlines (see section 4.2). In the J1 -domain (unirradiated area), d + -lines which could be expected
running from the corners to the centre along the bisection lines do not appear. Simultaneously,
in the J2 -domain (irradiated region), only a short d + -line exists running from the sample’s
corners to the boundary between the two regions. This is due to the fact that at T = 50 K the
unirradiated part of the sample is above the depinning line where the critical current is zero.
When the flux front has reached the d − -line at the narrowest point of the outer J2 -domain, which
separates the irradiated from the unirradiated zone, the vortices which are driven to the centre
of the J1 -domain pile up there and form a new flux front. This flux front moves oppositely
from the centre towards the d − -line as the J1 -domain is filled with flux lines. This flux front
then penetrates partly into the J2 -domain forming an additional d + -line in the J2 -domain (for
further details and field-cooling experiments, see Schuster et al (1994c)).
6.6. Domain boundaries with parallel currents: II
In this section we consider again current domain boundaries (d − -lines) separating areas of high
j2 and low critical current densities j1 , where the current is oriented parallel to the boundary
(perpendicular to n). Similarly to the situation which is described in the previous section, the
shape of the inner current domain corresponds to the sample geometry, but now the domain
of higher j2 is embedded in the j1 -region of the sample and thus magnetic flux first penetrates
the j1 -zone, and this is followed by a penetration into the j2 -domain during an increasing of
the external field. Additionally, the symmetry of the current pattern with respect to the sample
centre is broken due to a shift of the j2 -domain from the sample centre towards the right edge
of the square.
The region of larger j2 was prepared by a substrate modification-induced change of the
epitaxial growth of YBa2 Cu3 O7 films. The local variation of the substrate surface leads to
a modified microstructure in the superconducting film and thus to a variation in the vortex
pinning strength (Leonhardt 1998, Leonhardt et al 2000b). The substrate modification was
performed by irradiating a part of SrTiO3 substrates with a focused ion beam (FIB) of gallium
ions. By choosing the appropriate parameters of the FIB either an enhancement or a reduction
of the current-carrying capability of the superconductor in these areas can be obtained. This
offers a possibility of changing the value of the current density in defined areas of the samples.
For the preparation of a jc = j2 domain of high current density in an area of a YBCO film, a
square-shaped area of size 500 µm × 500 µm of the SrTiO3 surface was scanned with a FIB
of gallium ions. After the substrate treatment, a 130 nm thick YBCO film was deposited by
pulsed laser deposition and a square-shaped YBCO film was patterned by photolithography
with a lateral size of 1.5 × 1.5 mm2 (for more details see Leonhardt (1998) and Leonhardt et al
(2000b); for substrate irradiation with arrays of dots or lines, see also Leonhardt et al (2000a)).
In figure 44 the flux density distribution of the YBCO square with two current domains j2
and j1 is depicted. A square-shaped area of 0.5 × 0.5 mm2 with an increased jc = j2 is located
between the sample’s centre and the sample’s right edge. As regards the magnetic flux pattern,
it is difficult to recognize the jc -enhancement, since according to ∇ × j = µ0 B a jump in the
current density is related only to a discontinuity of the slope of Bz . A sharp bend of Bz (x, y)
can be seen in the two flux density profiles crossing the j2 -area. Additionally, the penetration
depth of the flux front penetrating from the sample’s right edge is slightly reduced. The current
domains which are separated by different discontinuity lines can be characterized as follows
(see figures 44 and 45): there are eight d − -lines, four of them representing the sample’s edges
and four of them separating the j1 -domain and the j2 -domain which are located within the
j1 -domain. All different kinds of d − -line can be seen in the flux density profile in figure 44 as
peaks, where the slope of the Bz (x) profile is discontinuously changing.
Current distributions in high-Tc superconductors
725
a)
b)
Figure 44. (a) Flux density distribution in a YBCO thin film with two different jc s at T = 5 K
after ZFC in an external field of µ0 Hex = 66 mT. The critical current density jc = j2 is enhanced
in a square-shaped area of 0.5 × 0.5 mm2 between the sample centre and the sample’s right edge
by a factor of j2 /j1 = 1.3. (b) A greyscale image of |j (x, y)| in a YBCO thin film at T = 5 K in
an external field of µ0 Hex = 66 mT. The YBCO grown on irradiated SrTiO3 exhibits a higher jc
as can be seen in the current density profile taken along the black horizontal line.
The current density profile in figure 44(b) shows a sharp transition from roughly j1 =
1.6 × 1011 A m−2 in the unmodified region to j2 = 1.9 × 1011 A m−2 in the modified region.
In addition to the current jumps at the inner d − -lines, the current density in figure 44(b) of j1
726
Ch Jooss et al
b)
a)
α
Figure 45. (a) A section of the current distribution of figure 44(b). (b) A sketch of the current
streamlines and discontinuity lines (bold) in and in the vicinity of the modified part of the
superconductor.
is continuously increasing towards the d − -lines of the sample’s edges, which could be induced
by calibration errors or electric field effects (see section 3.3 and 4.4).
In addition to the extra d − -lines, the d + -lines show a significant change in the area of the
j2 -domain. In contrast to the case for square-shaped samples with homogeneous jc , where
the d + -lines form an ‘X-like’ pattern, the d + -lines in figure 44 are tilted and split up in the
modified region. These lines indicate sharp bends of the currents as well as regions where flux
fronts meet. In contrast to the case of a domain of lower j1 being located in the sample’s centre
(see section 6.5), where the d + -lines disappear in the inner current domain, in figure 45 the
d + -lines continue from the j1 -domain into the j2 -domain. Due to the asymmetric position of
the modified area, the d + -lines do not continue straight. On the upper and lower border of the
modified region the d + -lines bend sharply. Furthermore, two extra d + -lines are visible which
end at the top and bottom edges of the modified area. These extra lines are not very high in
contrast, because the bending of the current at these additional lines is very small. On the side
of the sample left untreated within the j1 -domain, the black d + -lines seem to be undisturbed
by the presence of the j2 -domain.
Figure 45 shows a sketch of the discontinuity lines and the current streamlines in the
sample. The current streamlines have to fulfil the condition div j = 0. This leads to the
following relation between α, j1 and j2 :
α=
1
j1
arccos .
2
j2
(66)
In our case, α = 16.3◦ , which corresponds to the measured current densities of j2 =
1.9 × 1011 A m−2 and j1 = 1.6 × 1011 A m−2 . On the right side of the sketch, we find
discontinuity lines at the same positions as were found by measurement in figure 44(b). The
d + -lines which start at the left edges of the modified area are not visible in our measured data.
This is probably due to the small angle with which they propagate from the corner and thus to
the small enhancement of jc in the modified region.
7. Texture and local current distribution
Polycrystalline films and polycrystalline bulk ceramics of HTS exhibit critical transport
currents several orders of magnitude lower than those of highly textured materials such as
epitaxial thin films or single crystals. The decisive limiting factor for the currents in nontextured materials is the poor superconducting coupling between neighbour grains (Larbalestier
Current distributions in high-Tc superconductors
727
et al 1987, 1988, Ogale et al 1987, Peterson and Ekin 1988). This depression of the
intergranular critical current density flowing through grain boundaries (GBs) has been widely
documented in YBa2 Cu3 O7 thin film (Chaudhari et al 1988, Dimos et al 1988, 1990, Mannhart
et al 1988, Hilgenkamp et al 1996, Heinig et al 1999) and bulk bicrystals (Miller et al
1995, 1998, Field et al 1997, Jooss et al 2001a) and also for other HTS materials. For
Bi2 Sr2 Ca1 Cu2 O8 , see Zhu et al (1998); a review on GB research was written by Hilgenkamp
and Mannhart (2002). It was found that the critical current density through GBs is very sensitive
to the orientation of the adjacent crystallites and strongly decreases with the misorientation
angle θ for θ θ1 = 3◦ –5◦ (Heinig et al 1999). In particular, in YBa2 Cu3 O7 thin-film tilt
boundaries an exponential decrease jc ≈ jc,0 exp(−(θ − θ1 )/θc ) is observed with jc,0 ≈ 2–
5 × 1011 A m−2 at T = 4.2 K, self-field and θc ≈ 4◦ .
In efforts to reach large critical transport current densities in HTS materials, significant
progress was achieved due to the development of texturing techniques for HTS materials: in
bulk YBCO, different kinds of melt-texturing techniques (Jin et al 1988, Salama et al 1989,
Willén and Salama 1992, Cima et al 1992, Murakami et al 1990, 1992, 1993, McGinn et al
1990b, Gautier-Picard et al 1998) led to large single-domain samples (Lo et al 1997) with
critical current densities up to 6 × 109 A m−2 at T = 4.2 K (1 × 109 A m−2 at T = 77 K)
(Walter et al 2000, Jooss et al 2000c, 2001a). They exhibit a complex microstructure consisting
of peritectic particles, low-angle grain boundaries (LAGBs) and microcracks which are mainly
oriented parallel to the a–b plane.
For the development of superconducting tapes and coated conductors, different approaches
were taken. Tapes of Bi-2212 and Bi-2223 are produced by powder-in-tube techniques (for
a review see Vase et al (2000)). The crystallites in the Ag matrix display a certain amount
of alignment which is important for the current-carrying capability of the tapes. For the
development of YBCO-coated conductors on flexible polycrystalline substrates with a biaxial
alignment of the YBCO grains, two different approaches have been realized. Highly textured
YBCO films are produced either on rolling-assisted biaxially textured substrates (RABiTS)
(Goyal et al 1996, Norton et al 1998) or on polycrystalline metallic substrates with biaxially
textured buffer layers grown by ion-beam-assisted deposition (IBAD) (Iijima et al 1992,
2000, Freyhardt et al 1996). These YBCO films exhibit a network of LAGBs with an
angular distribution and a grain boundary length depending on the texturing process. For
an understanding of the transport properties of such GB networks one has to take percolative
aspects into account (Babic et al 1991, Asadov et al 1994, Evetts et al 1999).
This section is organized as follows. First we consider properties and, in particular, the
local current distribution of individual GBs in bicrystalline films and bulk samples (section 7.1).
In section 7.2 the intragranular and intergranular current densities of polycrystalline bulk
samples with a network of GBs are discussed. Section 7.3 focuses on the current-carrying
capability of melt-textured YBCO and finally the flux and current distributions of YBCOcoated conductors and multifilamentary Bi tapes are considered in sections 7.4 and 7.5,
respectively.
7.1. Grain boundaries
Since the current depression of GBs is the most serious problem for the development of HTS
applications, intensive studies of the microstructural, transport and electronic properties of
GBs have been performed over the past few years. The atomic structure of LAGBs consists
of an array of well separated GB dislocations (Chisholm and Smith 1989, Chisholm and
Pennycook 1991, Zandbergen et al 1990a, Browning et al 1993, Babcock 1999, Babcock and
Vargas 1995) accompanied with a strain field which decays over a characteristic length scale
728
Ch Jooss et al
Grain 1
a)
Grain 2
b)
θ
Figure 46. (a) A schematic model of a LAGB with separated dislocation cores. (b) A model of
a charged HAGB, where the electric charge is screened on a characteristic length scale le . This
leads to a bending of the electronic band structure in the neighbourhood of the GB (Schneider et al
1999).
ls ; see also the review of Jagannadham and Narayan (1994). In addition, the dislocations
may exhibit an electric charge distribution which decays over a length scale of the electronic
screening length le (see figure 46). At larger misorientation angles the GB dislocations start
to overlap and the atomic distortions at the GBs are no longer localized in dislocation cores.
The atomic structure in this case is built up from structural units which significantly deviate
from those of the bulk crystal structure (Browning et al 1998). The overall suppression of the
superconducting condensate in such GBs leads to a Josephson-junction-like behaviour of the
current-carrying capability, e.g. a Fraunhofer pattern can be measured for the magnetic field
dependence. These kinds of grain boundary are usually called high-angle grain boundaries
(HAGBs) in contrast to LAGBs. The critical angle separating the two regimes is usually
assumed to be θ ≈ 10◦ –15◦ . The critical currents in the LAGBs are of the same order of
magnitude or not more than one order of magnitude smaller than those of the adjacent grains.
Four different mechanisms are considered to be responsible for the current depression at
LAGBs:
(1) A deterioration of the superconducting state due to the strain field of the dislocation array
of the GB. This mechanism seems to be mainly important for not-too-large misorientations
θ, where the strain decay length ls is larger than the electronic screening length le (Gurevich
and Pashitskii 1998).
(2) If le > ls , a local Tc -depression due to a charge-carrier depletion created by the atomic
distortions of the GB dislocations could be dominant. The localized charge in the GB
dislocations leads to a bending of the electronic band structure on a length scale le
(Mannhart and Hilgenkamp 1997, 1998, 1999, Gurevich and Pashitskii 1998) (see figure
46). The impact of localized electronic states due to oxygen vacancies was emphasized
by Halbritter (1992), where the transport properties are described in terms of direct
and intermediate tunnelling processes. Oxygen deficiency in relation with structural
distortions at GBs was found by Zhu et al (1994).
(3) An order parameter suppression at the GB due to the different orientations of the dx 2 −y 2
order parameter in the adjacent grains (Hilgenkamp et al 1996, Mannhart and Hilgenkamp
1999). Since this effect gives a much weaker jc (θ ) than observed in the experiments, it is
not dominant and can be neglected in LAGBs (Gurevich and Pashitskii 1998).
(4) Quasiparticle scattering-induced order parameter suppression (Jooss et al 1999, 2000a).
Quasiparticle bound state were observed by Alff et al (1998). For details, see section 8.2.
Current distributions in high-Tc superconductors
729
Recent results of an enhancement of the intergranular jc by an order of magnitude caused
by overdoping of YBCO HAGBs proves that mechanism (2) plays an important role (Schneider
et al 1999, Schmehl et al 1999, Mannhart and Hilgenkamp 2000). The overdoping is performed
by partly replacing the Y atoms by Ca atoms. Since the ion radius of Ca is smaller than that of
Y, an additional contribution from the strain mechanism (1) cannot be ruled out. Recently, an
enhancement of jc by Ca doping was also obtained in LAGBs (Daniels et al 2000, Guth et al
2001)
To avoid the usage of specific critical state models for the analysis of integral measurements
such as transport measurements or magnetization measurements, it is necessary to obtain local
information on the grain boundary. This can be done by studying magnetic flux and current
distributions of extended grain boundaries or grain boundary networks by magneto-optics.
Magneto-optical measurements of grain boundaries can be performed on bicrystalline YBCO
thin films on bicrystalline SrTiO3 (for substrate properties see Jiang and Abell (1998)) or other
substrates, e.g. buffered bicrystalline Ni films (Thiele et al 2000). The YBCO films were either
grown by pulsed laser deposition or evaporated by sputtering techniques.
Investigating LAGBs with increasing misorientation angles θ , a drastic change in the
flux density distribution is found starting at angles above θ = 2◦ –3◦ . This indicates a strong
decrease of the critical current density jb across the LAGB with increasing θ in the range
3◦ < θ < 10◦ (Polyanskii et al 1996). In figure 47 the flux penetration into bicrystalline
YBCO films with a LAGB and a HAGB are compared. The most interesting feature is the
complete penetration of the HAGB with magnetic flux whereas no flux has penetrated the
adjacent films yet. The depicted flux density distribution is qualitatively indistinguishable
from that of two totally separated superconductors, indicating that the intergranular jc is very
small. In contrast, the LAGB is in a partly penetrated state; however, the penetration depth of
the flux along the GB is much larger than in the adjacent grains.
In the framework of Bean’s model with a constant intragranular critical current density
jc in the flux-penetrated part of the superconductor and a constant intergranular jb < jc , a
model for the current pattern near the grain boundary can be given (Polyanskii et al 1996).
A sketch of the current pattern for jb /jc ≈ 0.6 is shown in figure 47(c) and compared with a
measured flux and current pattern in figure 47(d). Typically there is a threefold bending of the
current streamlines around the GB: outside the GB two d + -lines are formed at an angle α to
the GB. An additional bending occurs at the GB itself. Here |j |, as well as ĵ , is changing. This
line can be considered to consist of two parallel d − -lines in the limit of vanishing distance.
The angle α of the d + -line is given by
1
jb
.
(67)
α = arccos
jc
2
Figure 47(d) shows a model-independent determination of the current distribution at a 4◦ [001]
tilt boundary obtained by inversion of the Biot–Savart law. A detailed analysis yields a nonconstant current density along LAGBs (Albrecht et al 2000a, Jooss et al 2001c) which depends
on the local magnetic flux density and displays a hysteretic behaviour during change of the
external magnetic field. For further details see Albrecht et al (2001); evidence for a fielddependent jb (B) due to vortex pinning by GB dislocations was found by Diáz et al (1998b)
and Jooss and Albrecht (2002). Hysteretic behaviour in HAGBs was seen by Kim et al (2000).
The condition that the flux line motion is directed perpendicular to the current streamlines (Lorentz force) and that vortices cannot cross d + -lines (see section 4.2) has drastic
consequences for the electric field distribution during flux penetration at grain boundaries: as
indicated in figure 47(c), all vortices which penetrate the region between the two d + -lines embedding the GB have to enter the sample at the intersection point of the GB with the sample’s
730
Ch Jooss et al
b)
a)
c)
α
d)
1 mm
Figure 47. (a) A greyscale image of the magnetic flux distribution of a square-shaped YBCO
thin film (d = 150 nm) with a 3◦ [001] tilt boundary after ZFC to T = 4.2 K and application
of a normal Hex . The film has a size of 1 mm2 . (b) The same, but with a HAGB (θ = 24◦ ).
(c) A sketch of the current pattern at a grain boundary for a ratio jb /jc ≈ 0.6, assuming Bean’s
model. The dashed lines with arrows indicate the direction of the magnetic flux movement during
flux penetration. α is the angle of the d + -line with respect to the GB according to equation (67).
(d) The magnetic flux distribution and overlaid current streamlines of part of a bicrystalline YBCO
thin film (d = 400 nm) with a 4◦ [001] tilt boundary after ZFC to T = 4.2 K and application of
µ0 Hex = 115 mT.
edge. Consequently, in a flux flow state during the sweep of Hex , the electric field at the GB
is drastically enlarged compared to that in regions outside the GB (see figure 48) and the characteristic timescales for flux flow and flux creep according to equation (12) are significantly
smaller in the GB than in the grain. This means also that for magnetization experiments jb
relaxes faster than jc , and in a relaxed flux creep state jb is measured at a smaller electric field
as compared to jc in the grain (Jooss et al 2001b). Figure 48(e) shows the minimum of E(x, y)
in a 3◦ [001] tilt boundary in a relaxed state of a magnetization experiment. This figure also
nicely shows the spatial dependence of E(x, y) from the flux front to the sample edges outside
the GB as proposed in figure 23. More experimental results for E(x, y) at planar defects are
given in Jooss and Albrecht (2002). The situation is different for transport measurements,
where even for small voltages at the contact pads, a high electric field is concentrated in the
GB and jb is measured at higher electric fields than jc (Diáz et al 1998a), see also Hogg et al
(2001). Direct experimental evidence for the suppression of the magnetic granularity by this
effect is found by Pashitskii et al (1995b) by comparing flux distributions for transport and
magnetization currents in granular Bi-2223 tapes.
In addition to the electric field distribution, the magnetic flux distribution influences the
measured jb -values. Magneto-optical measurements are usually made on samples of mm size
with long GBs. In this case, the magnetic stray field of the longitudinal currents (depending on
the GB length W ) strongly enhances the magnetic flux within the GB. This leads to a so-called
‘flux focusing’ effect, where the flux density in the GB is much larger than in the grain. This
Current distributions in high-Tc superconductors
731
Figure 48. (a) Time development of the electric field distribution in a magnetization experiment
at a GB. After ZFC, Hex is suddenly applied. The corresponding flux flow state decays after
some time into a flux creep state. (b) Part of the magnetic flux distribution and current streamlines
of bicrystalline YBCO films (4◦ [001] LAGB) after ZFC to 4.2 K and applying a perpendicular
external field of µ0 Hex = 65.6 mT. (c) A greyscale image of |E(x, y)| for the GB in (b) at t = 7.5 s
after application of µ0 Hex = 65.6 mT with a ramp rate of 40 mT s−1 . In the horizontal electric
field profiles, the position of the grain boundary (GB), the flux front (F) and a macroscopic defect
(D) are indicated with arrows. (d), (e) The same, but for a 3◦ [001] tilt boundary in a relaxed
magnetized state 10 s after application of µ0 Hex = 50 mT at 4.2 K.
732
Ch Jooss et al
can be seen in figure 47(b), where the local flux density inside the GB exceeds strongly the
applied external magnetic field. Transport measurements are usually done at microbridges of
widths of ≈10 µm and show jc s between 109 and 3 × 1010 A m−2 for 24◦ tilt boundaries at
T = 4.2 K (Hilgenkamp et al 1996). However, in magneto-optical images of thin films with
a θ = 24◦ [001] tilt boundary (figure 47(b)), jb is much smaller due to the high B and low Ec
and the sample seems to be divided into two separate parts.
In this section, only magneto-optical experiments on thin-film bicrystals have been
considered. For further investigations on melt-textured bicrystals, see section 7.3.
7.2. Polycrystals
We have analysed the properties of isolated grain boundaries; magneto-optical observation
of flux and current distributions inside polycrystalline bulk material will now be discussed.
A very high number of arbitrarily oriented grain boundaries can be found in sintered HTS.
Polycrystalline YBCO samples are usually prepared by sintering powders of Y2 O3 , BaCO3 and
CuO in a stoichiometric mixture. The microstructure is characterized by randomly oriented
grains with sizes of about 5–40 µm depending on the sintering procedure (see e.g. Schuster
et al (1992a)). Most of them are separated by HAGBs, where the misorientation angle θ of
neighbouring grains is larger than 15◦ . Sometimes, the existence of amorphous intergrain layers
of YBCO with thicknesses of a few nm between the grains can be detected by transmission
electron microscopy (TEM) (Laval and Swiatnicki 1994). The wide-ranging variation
of the properties of the grain boundaries leads to non-trivial behaviour of polycrystalline
superconductors, depending on the magnetic history and the external field value. A model
of Forkl (1993) which reflects the magneto-optical observations describes the magnetic flux
penetration after ZFC in three different steps. Therefore, it is necessary to define a pair of
lower critical fields, Hc1,b for the intergranular phase and Hc1 for the single-crystalline grains.
For Hex Hc1,b , Hc1 , the superconductor is able to screen the magnetic flux completely out
of the sample. This is due to a macroscopic Meissner current density jM which is flowing near
the surfaces of the entire sample. For Hc1,b Hex Hc1 , magnetic flux penetrates into the
GBs (and, if present, into the intergranular phases). Due to the higher value of the intragranular
jM , the individual grains remain flux free. For H > Hc1 , flux starts to penetrate also into the
grains. However, due to the angular dependence of Hc1 (θ ) the first flux penetration into the
grains occurs over a broad range of values.
This can be seen in the magneto-optical image of a polycrystalline YBCO sample in
figure 49. Clearly visible is the granular structure inside the sample. Bright parts indicate
regions which are fully penetrated by magnetic flux; that means that the local lower critical
field is exceeded, and magnetic flux has penetrated into the superconductor. The dark areas
refer to parts of the sample which screen the external magnetic flux completely. Most of
them are single grains with orientation of the crystallographic c-axis parallel to the external
field. Note that flux-filled parts are interspersed in the whole sample. In contrast to the case
for textured or single-crystalline samples, neither a macroscopic flux density gradient nor a
connected flux-free area in the centre the superconductor can be detected.
It is remarkable that a 2D inversion of the Biot–Savart law can also reveal significant
information on the current distribution of samples with 3D complex current patterns (Albrecht
et al 1998) (see figures 49(b) and (c)). This is due to the fact that magneto-optical imaging
together with the inversion acts like a microscope with finite depth of focus which is mainly
sensitive to the currents in a surface layer of ≈20 µm close to the MOL. Figure 49(b) shows
the 2D current density distribution |j (x, y)| as a greyscale image. The white parts refer to
the highest current densities of 8 × 109 A m−2 . Remarkably, these local current densities are
Current distributions in high-Tc superconductors
733
Figure 49. (a) A greyscale image of the normal component Bz (x, y) at the polished surface
of polycrystalline YBCO after ZFC to 5 K and µ0 Hex = 64 mT. (b) A greyscale image of the
absolute value of the current density distribution |j (x, y)| calculated from the flux density in
figure 49. The greyscale variation from black to white represents current densities between 107
and 8 × 109 A m−2 . (c) Superposition of the microstructure and current streamlines. Different
greyscales refer to different grain orientations. The black area on the right side of the images is a
large pore from which the superconductor is screened by intergranular currents.
three orders of magnitude larger than possible transport currents in polycrystalline YBCO. The
length scale of the spatial variation of the current density corresponds to the grain sizes of the
sintered material which are in the range of 15–40 µm for the specimen considered.
A macroscopic intergranular screening current along the sample border which is known of
for a homogeneous superconductor is not observed. Because of the large number of arbitrarily
oriented grain boundaries along the course of such a global screening current, its current density
is suppressed below 107 A m−2 and therefore not detectable in this measurement. The current
distribution exhibits a strong correlation with the local grain configuration. In figure 49(c),
the current streamlines of a part of the sample overlie a polarization light micrograph of the
surface of the superconductor. Different greyscales refer here to different crystallographic
orientations of the grains. The grain boundaries are responsible for the local suppression of jc
and thus closed loops of circulating magnetization currents are formed. However, the degree
of localization of the current loops within single grains differs significantly. Regarding the
large grain in the lower right corner, the closed current loops are completely located within the
grain. The current streamlines are flowing parallel to the grain borders, and consequently the
currents across the boundaries nearly vanish. In contrast to this behaviour, there are several
grain boundaries, e.g. in the top area of figure 49(c), which do not significantly disturb the
local current flow. In this case, either the GB angle is smaller or the alignment of the grains
734
Ch Jooss et al
(c)
Figure 50. Left: (a) a schematic diagram of the model sample, consisting of a hexagonally closepacked lattice of YBCO discs with 2r = 50 µm and contact width w = 3 µm; (b) polarized light
images of a part of the sample; (c) magnetization loops of the model sample at T = 5, 10, 15, 20,
30, 40, 50 and 70 K. The dashed lines indicate the peak positions. Right: the flux distribution after
ZFC to T = 18 K and application of µ0 Hex = 18 mT (a), 45 mT (b), reduction to 39 mT (c) and
the remanent state (0 mT) (d); due to demagnetization effects and the artificial granularity of the
sample, negative vortices penetrate at the weak links and small annihilation zones are formed at
each individual disc (Koblischka et al 1999a).
is much better. Thus, a number of grains form an area which is screened collectively by the
flowing supercurrents and the local critical current density of such areas is relatively large.
A question which is deeply related to the flux and current distribution of granular HTS
superconductors is that of the observation of complex and asymmetric magnetization curves
which strongly deviate from the expectation of the Bean model. As a result of the magnetic
field dependence of jc (B), hard type-II superconductors always exhibit a low-field peak in
the magnetization curve in decreasing Hex . In the framework of the Bean model the peak is
expected to be located at zero or at slightly negative Hex (Shantsev et al (1999) and references
therein); examples of theoretical calculations are given by Brandt and Indenbom (1993b),
McDonald and Clem (1996); experimental data can be found in Senoussi (1992). Recently,
magneto-optical investigations of an artificially granular YBCO film (representing a thin-film
model system designed to reproduce the behaviour of polycrystals) have revealed clearly how
the shift of the magnetization peak towards positive fields is caused by percolative granular
flux and current patterns (Koblischka et al 1999a, Shantsev et al 1999); see figure 50.
Current distributions in high-Tc superconductors
735
In a decreasing external field, the magnetic flux first starts to move out of the granular
sample in the weak-link network. In contrast to the case for single-crystalline films (see e.g.
figure 20), the area of reversed flux gradients and current densities does not grow only from the
sample’s edges. Depending on the current suppression in the weak links, flux exit takes place
over the whole sample, starting at the surfaces of the individual grains. Due to the inverse strayfield peak of the reversed currents at the grain edges, negative flux starts to penetrate the grains
if the remanent state is reached. Vortices of opposite sign penetrate the sample and annihilate
with the trapped vortices. Consequently, at the remanent state, a significant amount of negative
flux is already present in the sample and a minimum of the vortex density in decreasing Hex
occurs before the remanent state is reached. This behaviour can be nicely seen in the experiment
of Koblischka et al (1999a). The remanent state of the artificial granular sample is depicted in
the right column of figure 50 in (d). The reason for the magnetization peak at positive fields in
granular samples is then explained by the minimum flux density at Hex > 0 in decreasing fields.
For a theoretical modelling of the flux pattern in granular square-shaped thin films
containing a Josephson network in increasing and decreasing fields, see Reinel et al (1995).
7.3. Melt-textured material
The strongly suppressed transport current density in ordinary polycrystalline samples has
always been a limiting factor for the application of bulk superconductors. Therefore a melttexturing technique for the production of bulk HTS has been developed in order to eliminate
the HAGBs inside the sample (Jin et al 1988, Salama et al 1989). YBCO powder is heated
beyond its peritectic point of T ≈ 1320 K where it melts incongruently into a solid Y2 BaCuO5
phase and a liquid phase rich in Ba and Cu.
On cooling the semi-solid melt very slowly under the peritectic point the 1–2–3 phase is
formed. The microstructure is dominated by long plate-shaped grains aligned over wide areas.
Jin et al (1988) achieved critical current densities of jc = 1.7 × 108 A m−2 at 77 K in zero
field, which is about two orders of magnitude higher than for as-sintered material. Similar
results were obtained by McGinn et al (1990a and b) by using a zone-melting method, where
jc -values up to 5 × 108 A m−2 at 77 K are achieved (McGinn et al 1990a).
An improvement of the mechanical as well as of the transport properties in melt-textured
YBCO can be achieved by using non-stoichiometric precursors, e.g. with Y2 BaCuO5 (Y211) or
Y2 O3 additions, additions of Pt or CeO2 (Ullrich et al 1999, Leenders et al 1999), Ag (Mendoza
et al 2000) and by applying a top-seeded melt growth process. This leads to samples with a high
density of Y211 particles with sizes of less than 1 µm which are favourable for flux pinning and
reduce the number of microcracks. Recently, a jc of 1×109 A m−2 at 77 K (jc = 6×109 A m−2
at 4.2 K) was achieved in this material (Walter et al 2000, Jooss et al 2001a). Figure 51 shows
a polarized light micrograph of an a–c surface of such a melt-textured YBCO sample. The
number of large cracks is significantly reduced and small Y211 particles lead to a hardening
of the material and contribute to flux pinning (Martı́nez et al 1996). The improvement of the
mechanical stability is a big issue, since the Lorentz forces in these materials are already large
enough to destroy the bulk samples in magnetization or transport current experiments.
The growth of relatively large highly textured domains and the reduced appearance
of precipitates between neighbouring grains in the melt-textured samples leads to a totally
different behaviour in an external magnetic field compared to that of the sintered polycrystalline
material. The description by different steps of penetration is not valid in this case. Magnetooptical studies of the current-limiting structures in YBCO melt-textured material were made by
several authors (Vlasko-Vlasov et al 1994b, Hedderich et al 1995, Uspenskaya et al 1997, Jooss
et al 2000c, 2001a, Walter et al 2001). The magnetic flux penetration in melt-grown NdBaCuO
736
Ch Jooss et al
Figure 51. A scanning electron micrograph of an a–c surface of currently available optimized
melt-textured YBCO containing small Y211 particles (Leenders 2000). Microcracks within the
a–b plane and pores are also visible.
(c)
Figure 52. Left: polarized light images of two faces, (a) c and (b) (a, b) of a multidomain meltprocessed YBCO sample. (c) A schematic diagram of the orientation of the individual domains.
Right: magnetic flux patterns after ZFC to 16 K and application of µ0 Hex = 93.5 mT (a), (b) and
a remanent state after µ0 Hex,max = 117 mT (c), (d) (Uspenskaya et al 1997).
material in the a–b and a–c planes was visualized by Zamboni et al (1998) and for different
rare-earth-substituted REBaCuO by Koblischka et al (1998a) and Zamboni et al (2000).
Figure 52 shows magnetic flux patterns of a multidomain melt-textured YBCO sample at
different surfaces of the rectangular bar. The easy flux penetration in increasing and flux exit
with decreasing Hex indicates the positions of the grain boundaries. Profiles of the flux density
distributions across two differently oriented domains are shown in figure 53. In their study of
thick samples, Uspenskaya et al (1997) take into account the lens-shaped flux penetration (see
section 4.6) for the determination of the current density from the measured flux profiles. The
Current distributions in high-Tc superconductors
737
a)
b)
c)
Figure 53. (a) Profiles of the perpendicular flux components across domain 2 (see figure 52) at
15 K. The lowest curve was obtained after ZFC and application of µ0 Hex = 20 mT; for each
subsequent curve µ0 Hex was increased by 10 mT. (b) As (a), but for domain 1, where the c-axis
lies perpendicular to z and x. (c) The anisotropy ratio of the critical currents jc,ab (in the a–b plane)
and jc,c (parallel to the c-axis) (Uspenskaya et al 1997).
anisotropy ratio of the critical current density jc,ab (flowing in the a–b plane) to jc,c (flowing
parallel to the c-axis) is determined as 2.5. As can be seen in figure 53(c), this ratio strongly
increases above temperatures of 60 K, which reflects the anisotropy of the irreversibility line
in this material.
A serious problem that limits the possible jc,c even in single-domain melt-textured YBCO
is the high number of microcracks (a, b) in the samples which are formed during the
processing due to the anisotropic thermal expansion and the tetragonal–orthorhombic phase
transition during the oxygen loading. Figure 54 shows the microcracks in high-resolution
magneto-optical images. The microstructure is visible in the polarized light images of a a–
c surface of the melt-textured YBCO sample (Schuster et al 1993b). The anisotropic flux
penetration is both due to the intrinsic anisotropy of the critical current densities jc,ab and
jc,c and due to the microcracks in the a–b plane. Particularly striking is the filamentary
pattern of easy flux penetration along the microcracks in the a–b plane. They represent weak
links which separate bulk melt-textured samples into plates of relatively large jc . However,
for current transport parallel to the a–b planes their influence is probably not very large,
since their extent in the a–b plane is usually much smaller than the sample size in the same
direction.
Figure 55 shows the flux and current density distributions in the a–b plane of an optimized
melt-textured YBCO strip (prepared with precursors of Y-123 + 25 mol% Y2 O3 + 1 wt%
738
Ch Jooss et al
Figure 54. (a) A polarized light image of a a–c surface of a melt-textured YBa2 Cu3 O7 sample. (b)
A section of (a) showing the platelike microstructure with a high number of microcracks oriented
in the a–b plane. Also shown is the flux density distribution measured with a MOL of EuSe after
ZFC to 5 K and applying µ0 Hex = 256 mT (c) and 512 mT (d). Bright parts on the left and on
the right sides of the sample indicate an easier flux penetration in the a–b direction than in the
c-direction. Note the preferential flux penetration along the microcracks.
a)
b)
Figure 55. (a) Magnetic flux distribution together with current streamlines and a current density
profile of a strip of melt-textured YBCO at T = 80 K. An external magnetic field of 400 mT
was applied and then reduced to µ0 Hex = 56 mT. (b) A greyscale representation of the absolute
value |j (x, y)| of the current density distribution for the strip. Some bright spots correspond to
distortions of the magneto-optical layer.
Current distributions in high-Tc superconductors
739
a)
I
c)
ds
de
II
GB
1 mm
III
α
c
a,b
b)
a,b-growth
z ab
c-growth
c
θ
zc
a,b
adsorbants / pellets
GB
1 mm
Figure 56. An optical image (a) of a multiply top-seeded melt-textured YBCO sample (parallel to
the a–b plane) containing an impure 9◦ [001] tilt boundary and (b) the magneto-optically observed
flux distribution at T = 4.2 K in the remanent state after application of µ0 Hex = 176 mT parallel
to the c-axis. (c) A sketch of the growth domains and growth directions of a double-seeded sample
which is cut parallel to c near the seeds. Different growth domains occur at the θ [001] tilt boundary:
the a–b growth domains, where the GB becomes highly non-stoichiometric, and the c-direction
growth domain, where clean GB can be obtained.
CeO2 (Leenders et al 1999)) at a higher temperature (T = 80 K) and after application of
µ0 Hex ≈ 400 mT. The average critical current density is jc ≈ 2 × 108 A m−2 ; fluctuations of
jc occur due the varying density and size of 211 particles. Interestingly, a maximum current
density occurs in a large part of the central zone of the strip, just before the current density
changes its sign. This maximum is related to a maximum trapped flux density of ≈180 mT
and indicates an increasing jc (B) with increasing local flux density, as often observed in
magnetization measurements on this material (the so-called fish-tail effect).
Much progress has been achieved over the past few years in growing large single-domain
YBa2 Cu3 O7 samples with sizes up to 7 cm by top-seeded melt-textured growth (TSMG).
However, it is known that the intragranular jc degrades with increasing distance from the seed
(Lo et al 1997). Consequently, larger samples have to be formed by applying a multiple topseeded melt growth (MSMG) technique or by joining single domains. Since this is unavoidably
related to the formation of GBs, their properties in melt-textured material are of current interest
(Field et al 1997, Miller et al 1995, 1998, Jooss et al 2001a). In addition to the intrinsic atomic
structure (see section 7.1), jb through GBs in melt-textured YBCO can be suppressed by nonstoichiometic accumulation of a Cu-rich liquid phase (Ekin et al 1987, Delamare et al 2000)
or of Y211 particles (Delamare et al 1999) at the GB due an incomplete reaction during the
growth process. However, there has been significant progress in growing clean GBs also in
melt-textured YBCO bicrystals (Willén and Salama 1992, Todt et al 1996, Jooss et al 2001a).
In order to show the differences between clean and non-stoichiometric GBs in melttextured monoliths and also the tilt boundaries in YBCO films, we discuss here magnetooptical experiments on an impure 9◦ and a clean 11◦ [001] tilt boundary which were formed
by a multiple-seeding melt growth method (Jooss et al 2000c, 2001a). Figure 56 shows an
optical micrograph and the flux distribution of a sample with a 9◦ [001] tilt boundary. The
grain boundary plane shows three different orientations on a macroscopic scale. Whereas
plane II is very porous and exhibits a large amount of normal phase, plane I is almost clean.
The large amount of porosity and impurity in plane II is due to the oppositely propagating a–b
740
Ch Jooss et al
growth fronts of the adjacent grains during the melt-texturing process (see figure 56(c)). In
contrast, at plane I a non-vanishing a–b growth component parallel to the boundary is present
which pushes the excess liquid and impurities parallel to the boundary and the concentration
of non-superconducting phases is much lower. This is reflected also in the magneto-optical
image in figure 56(b). In the right part of the image which corresponds to the outside border of
the sample one observes an overall depletion of the critical current density which is reflected by
the larger penetration depth of the magnetic flux density. This is due to an overall degradation
of the sample towards the outside border and thus jc and jb are lower in this region. The central
part of plane II exhibits a large zone of penetrated flux due to the pores and normal phases with
a width up to 400 µm. The best flux screening is obtained in the clean part (plane I) of the
boundary and in the right part of plane II where the amount of normal phase is significantly
lower and the grain boundary has some clean zones.
Figure 57 shows the flux and current distribution of a 11◦ [001] tilt boundary where the
degree of non-stoichiometry was drastically reduced by reducing the seed distance (Jooss
et al 2001a). At µ0 Hex = 16 mT the GB is completely screened from the external field
(figure 57(a)). However, magnetic flux is already penetrating some of the cracks parallel to
the a–b plane. On further increasing the external field up to 129 mT the magnetic flux only
penetrates into the a–b growth region of the GB. It is necessary to apply an external field of
≈350 mT to reach a state where magnetic flux completely enters the GB in the c-direction
growth domain. Figure 57(d) shows the distribution of the absolute value of the current density
after full flux penetration into the GB.
The complex current distributions in figures 57(c) and (d) reflect different properties of
the MSMG samples parallel to the c-axis: shielding and critical intragranular currents are
present not only parallel to the GB, they also shield the magnetic flux which penetrates into
the a–b plane cracks and give rise to a stripelike current distribution. In addition, the intrinsic
anisotropy of the superconducting properties parallel to c and parallel to the a–b plane is
reflected in different intragranular values: jc,ab = 2 × 109 A m−2 and jc,c = 9 × 108 A m−2 .
The intergranular critical current density though the 11◦ tilt boundary shows values up to
3 × 108 A m−2 in the a–b growth domain and up to 6 × 108 A m−2 in the c-direction growth
domain, which is only a third of the bulk jc,ab .
In addition to multiple-seeded melt-textured growth, large bulk samples of complex shape
can also be formed by joining of YBCO monoliths near the peritectic temperature under external
pressure. Here, large intergranular critical current densities up to 109 A m−2 at 55 K (400 mT)
(Walter et al 2001) and up to 2 × 108 A m−2 at 77 K in a self-field are obtained (Delamare
et al 2001). The local current density through joined YBCO monoliths was magneto-optically
visualized by different groups (Zheng et al 1998, 1999, Kawano et al 2000a, Jooss et al 2000c,
Claus et al 2001).
We have seen so far that the density and the misorientation angles of grain boundaries
in both HTS bulk material and thin films influence the current pattern very strongly. The
application of techniques such as the melt-texturing processes can significantly reduce the
limiting effect of the grain boundaries on transport currents. This enables bulk superconductors
now to be used in large-scale applications such as superconducting bearings and magnets or
superconducting flywheel energy storage systems, where high trapped fields and thus large
current densities in a bulk material are required (Leenders et al 2001). The following sections
report on the local current-carrying capability of superconducting tapes and wires.
Current distributions in high-Tc superconductors
741
Figure 57. Magnetic flux density and current density distributions of a melt-textured bicrystal with
a clean 11◦ [001] tilt boundary at an a–c plane. (a), (b) The magnetic flux density distribution
Bz (x, y) after ZFC to 4.2 K at µ0 Hex = 16 mT (a) and 129 mT (b). (c), (d) Greyscale images
of |j (x, y)| at the surface of the sample as obtained by inversion of the Biot–Savart law. (c)
corresponds to the Bz (x, y) of (b) and (d) shows the current density with µ0 Hex = 117 mT,
decreasing after application of 400 mT. (e) A backscattered electron image of the same section of
the sample with the GB (broken line) and the boundary between the c-direction and a–b growth
domains (bold lines). (f ) Energy-dispersive mapping of the Y concentration reflects the different
contents of Y211 in the two growth domains.
7.4. YBCO-coated conductors
After having considered the current density distribution and current-limiting factors in YBCO
bulk materials, the behaviour of networks of grain boundaries in YBCO-coated conductors
(CC) will be analysed. There are two different basic architectures of YBCO CC, where the
texturing process is either applied to the metallic substrate or applied to the ceramic buffer
742
Ch Jooss et al
Figure 58. Flux patterns for IBAD-YSZ/TCE-YBCO conductors with different FWHM of in-plane
texture: (a) 7◦ , (b) 13◦ and (c) 22◦ , after ZFC to 4.2 K and application of µ0 Hex = 32 mT. The
thickness of all films is 500 nm.
layer. (i) CC based on highly textured yttrium-stabilized ZrO2 (YSZ) buffer layers (Iijima
et al 1992, 2000, Freyhardt et al 1996). They are sputtered by ion-beam-assisted deposition
(IBAD) on polycrystalline Ni, stainless steel or YSZ substrates. (ii) CC based on rollingassisted biaxially textured substrates (RABiTS), such as Ni or Ni alloys (Goyal et al 1996,
Norton et al 1998, De Boer et al 1999) with epitaxial YSZ/CeO2 buffer layers. The YBCO
films are grown epitaxially on the biaxially textured buffer layers which exhibit a finite degree
of texture of full width at half-maximum (FWHM) up to 4◦ (out of plane, deduced from a (100)
rocking curve) and up to 7◦ (in the plane, deduced from (111) φ-scans). They are deposited by
different methods such as pulsed laser deposition (PLD) and thermal coevaporation (TCE) or
chemical methods like sol–gel or chemical vapour deposition (CVD). Due to the finite degree
of texture, these conductors contain a high number of LAGBs which significantly determine
the current-carrying capability of the tapes.
Since the YBCO films grow epitaxially, the texture is determined by the surface texture
of the underlying buffer layer. For IBAD layers, the alignment is improved with increasing
film thickness of the buffer layer. Therefore, the critical current density of the YBCO layers,
which depends exponentially on the degree of texture, can be simply varied by changing the
thickness of the IBAD buffer. Figure 58 shows a series of IBAD-YSZ/TCE-YBCO conductors
with the same thickness d = 500 nm and different in-plane textures of φf whm = 7◦ , 13◦ and
22◦ . To all of them the same external field of 32 mT was applied after ZFC, and the different
critical current densities are directly visible from the increasing penetration depth of the flux
front for increasing φf whm .
All IBAD conductors in figure 58 exhibit a sharply defined flux front separating the Meissner phase from the inhomogeneous flux-penetrated Shubnikov phase. On the length scale of
the spatial resolution of the experiment of 5.9 µm, no granularity can be seen; however, the
Current distributions in high-Tc superconductors
743
Figure 59. Magnetic flux penetration at T = 4.2 K in networks of GBs in different types of YBCOcoated conductor: (a) IBAD-YSZ/PLD-YBCO conductor (dYBCO = 300 nm) with φf whm = 25◦
at Bex = 4.8 mT, (b) IBAD-YSZ/TCE-YBCO conductor (dYBCO = 600 nm) with φf whm = 8◦
at Bex = 16 mT and (c) Ni RABiTS/CeO2 /YSZ/TCE-YBCO conductor (dYBCO = 600 nm) with
φf whm = 8◦ at Bex = 80 mT.
Figure 60. (a) A high-resolution magneto-optical image of the flux pattern of the IBAD conductor
A visible in figure 59(a). The greyscale variation covers a range from Bz = 0.05 mT to Bz = 1 mT.
(b) A planar view TEM image of the grain structure of the same sample as in (a). Grains and twin
boundaries are visible. (c) A TEM image of individual grains separated by LAGBs.
flux pattern in figure 58(c) is more blurred compared to those for the two other samples. The
same can be seen in figures 59(a) and (b), where the magnetic flux patterns of three different
types of CC containing a network of GBs in a partly penetrated state are compared (Jooss et al
2001c). The IBAD conductors in figures 59(a) and (b) exhibit a sharply defined flux front separating the Meissner phase from the inhomogeneous flux-penetrated Shubnikov phase (which
is however more blurred for the sample with φf whm = 25◦ (figure 59(a))). For small local
magnetic flux densities the magnetic flux shows a diffuse granular pattern which is shown in
figure 60(a) at a higher magnification. In contrast, in figure 59(b) no granular flux pattern is
visible and the roughening of the flux front is similar to that for single-crystalline films, where
it has its origin in small local variations in jc , small precipitates of Y2 O3 of a size <1 µm and
rough, crooked sample edges.
A completely different behaviour is observed for the flux pattern in the RABiTS conductor
with φf whm ≈ 8◦ in figure 59(c). The flux pattern shows long flux filaments with a length up
to 100 µm. A unique flux front oriented parallel to the sample edges as observed in the IBAD
conductor with similar texture in figure 59(b) does not appear. For Bex = 80 mT magnetic
flux has already penetrated into the sample centre along some GBs, whereas large areas which
correspond to individual grains remain completely flux free.
Figure 61 correlates the magneto-optically observed flux pattern in a Ni RABiTS conductor
with the GB location in the YBCO film and in the Ni substrate (Feldmann et al 2000). In contrast
744
Ch Jooss et al
Figure 61. Top: magneto-optical images taken across the whole width of a 4 × 10 mm YBCO
RABiTS conductor (in-plane FWHM ≈6◦ –8◦ , d = 600 nm) after ZFC to 15 K and applying
µ0 Hex = 60 mT. Two straight line defects on the left derive from scratches on the substrate.
Bottom: a magneto-optical image (a) and light microscopy images (b) of the YBCO surface and
(c) of the underlying Ni substrate after the buffer layers were etched away. All images are taken
at the same location. (d) Parts (a) and (c) superimposed, showing that preferential flux penetration
occurs almost only along the Ni GBs and that not all Ni GBs admit flux (Feldmann et al 2000).
to the case for IBAD conductors, where the magnetic grain size reflects the size of the YBCO
growth islands, the magnetic grain size of the RABiTS conductors is determined by the grain
size of the metallic substrate. Since the magnitude of preferential flux penetration at a GB
increases with decreasing jb /jc , a qualitative comparison of the current-carrying capability
of the GBs and a correlation with the misorientation angle is possible. This was done by
Feldmann et al (2000) and is shown in figure 62. The best coincidence between the magnetooptically observed flux pattern and the location map of the Ni grain boundaries is obtained for
a threshold angle θ 4◦ .
The montage of figure 62 focuses attention on the factors that control the minimum
misorientation angle at which LAGBs become a barrier to the current flow. Transport
measurements suggest that this takes place in the vicinity of 7◦ for [001] tilt boundaries
(Turchinskaya et al 1994, Heinig et al 1999), but other studies suggest angles near 4◦ –5◦ .
Current distributions in high-Tc superconductors
745
(g)
Figure 62. Top: a magneto-optical image (c) and the corresponding backscattered electron Kikuchi
pattern (BEKP). (a) A BEKP percolation map in which all points have the same greyscale level
if the point-to-point misorientation is 1◦ . (b) As (a), but with a criterion of 4◦ . (d)–(f ) GB
maps of the location of all GBs with θ 1◦ , 4◦ and 7◦ , respectively. The 4◦ boundary map in
(e) strongly resembles the MO image in (c). Bottom: flux density in the GBs as a function of GB
misorientation θ from figure 61. Below a threshold angle of 4◦ no preferential flux penetration is
visible (Feldmann et al 2000).
At present, it is not fully clear whether the threshold angle in CC is different because the
GBs are not pure [001] tilted; they are more grooved or their behaviour is different because
the intragranular current density is higher (perhaps due to a larger defect density). It should
746
Ch Jooss et al
be noted that magneto-optical determination of jb in magnetization experiments could result
in smaller threshold angles than from transport measurements due to the lower electric field
criteria.
For a direct comparison of the YBCO films on IBAD and the RABiTS substrates, the
differences in grain size and, thus, in length of the GBs has to be taken into account. The
typical grain size in the RABiTS conductor in figure 59(c) is 30–50 µm. Transmission electron
microscope images of the grain structure of the IBAD conductor in figure 59(a) are presented
in figures 60(b) and (c). The grains visible in the image have a size of 100–300 nm. A detailed
analysis of individual grains in figure 60(c) shows that larger grains still contain LAGBs. Thus,
the average grain size is 100 nm rather than 300 nm and the grain size is approximately 300
times smaller compared to that for RABiTS conductors. Different grain boundary lengths lead
to different length scales of the current percolation and consequently to different length scales
of a possible affinity of the magnetic flux patterns in the two types of sample. Figure 60(a)
shows a high-resolution image of the magnetic flux pattern of figure 59(a) for an area of
30 × 30 µm2 . It displays a granular structure on length scales of about 2–3 µm, indicating
that several grains with a high alignment are screened collectively by the flowing Meissner
currents.
Summarizing this section, the GBs are clearly identified as representing the currentlimiting factors in the YBCO-coated conductors. However, it is not completely clear at the
moment whether the GBs in these materials are identical to those in thin-film bicrystals, since
mixed tilt axes, grooving or separation of GB dislocations into partial dislocations could give
rise to differences in the current-carrying capability. The consequences of a grain size of
≈100 nm in PLD YBCO on IBAD buffer layers, such as a depletion of the intragranular
current density or differences in the pinning of vortices in the GB planes, are not understood
at present.
7.5. Bi-2212 and Bi-2223 tapes
The powder-in-tube (PIT) technique applied on the (Bi, Pb)2 Sr2 Ca2 Cu3 O10+δ (Bi-2223)
compound with a Tc of 110 K and to some extent also to Bi2 Sr2 CaCu2 Ox (Bi-2212) with
a lower Tc of 85 K has been the most successful technique for making long HTS up to now
(Pabst et al 1995, Vase et al 2000). For a comparison of different materials and approaches,
see Larbalestier et al (2001). Even if their maximum jc of (5–10) ×108 A m−2 at 77 K
is much lower compared to that for the YBCO CC (jc of (1–3) ×1010 A m−2 at 77 K),
this disadvantage may be compensated by larger cross sections of the conductor, in order to
obtain large total critical currents Ic . In contrast to the case for the CC, where only 10 m
lengths are available at present, km lengths of Bi-2223 tape with jc above 1.5 × 108 A m−2
are successfully manufactured (Malozemoff et al 1999, Fischer et al 2001). This opens up
the possibility of fabricating several materials suitable for applications such as high-current
wires and superconducting coils. However, for the YBCO CC also, which exhibit much better
magnetic field performance, enormous efforts are being made at present to achieve an upscaling
to lengths of hundreds of metres and kilometres.
In the PIT technique, the ceramic superconducting powder is placed into a silver alloy
tube in order to allow mechanical processing, like swaging, drawing, rolling and pressing of
the brittle ceramic material. There are of course variations in the manufacturing process from
company to company, but the overall concept is the same. The precursor powder for Bi-2223
tapes typically consists of Bi-2212 and secondary phases, which later in the thermomechanical
process is converted to the Bi-2223 phase. The precursor-filled tube is then drawn down to
transform the short straight tube into a flexible single-filament wire. The phase conversion
Current distributions in high-Tc superconductors
747
c)
d)
Figure 63. (a), (b) Features of transport behaviour of individual high-jc filaments for currents
applied parallel to the long filament axis which show the contribution of c-axis transport to currents
in the a–b plane. (a) V –I curves measured at 2, 4, 6, 8, and 10 T; inset: at self-field. (b) The
temperature dependence of Ic at 8 T, typical for the c-axis contribution to Ic and showing the
presence of GBs. (d) A plan-view magneto-optical image of an 85-filament Ag-sheathed Bi-2223
tape after field cooling in a field of 40 mT to 16 K and then switching off the field. Vertical flux
structures reflect the filament structure of the tape. Quasiperiodic horizontal lines of easy flux
penetration indicate unhealed cracks crossing the filaments. The image (c) shows a cross section
of the tape (Cai et al 1998).
during heat treatments plays an important role in the development of texture: the degree of grain
alignment is much better than for the texture induced by mechanical deformation alone. In
addition, the liquid phase is important for healing cracks which are created during mechanical
deformation. Ag is chosen as the matrix material for chemical reasons: it does not react with
the precursor and allows oxidization during the heat treatment.
Since many high-current applications are operated with alternating currents, the ac losses
of the wires can be reduced by decreasing the width of the individual wire cores and placing
several of them into a new Ag tube in order to make multifilament wires. Several configurations
of multifilament and monocore tapes of Bi-2223 and Bi-2212 have been developed during the
past few years.
There is much evidence that the upper limit for jc in Bi tapes is not already reached
(Cai et al 1998, Larbalestier et al 1999). Thin-film samples of Bi-2223 exhibit jc in the
range of 109 –1010 A m−2 at 77 K and the self-field (Wagner et al 1995) and 1011 A m−2 at
20 K (Rössler et al 2001); the current-limiting mechanisms of the Bi tapes (exhibiting a very
complex microstructure) are still not fully clear (Larbalestier et al 1999). Magneto-optical
experiments revealed that the current flow percolates in the Bi tapes (see e.g. (Nakamura et al
1992, Welp et al 1995a, 1995b, Pashitskii et al 1995a, 1995b, Polak et al 1997b, Zhang et al
1998, Jiang et al 1999, Kawano et al 2000b, Polyanskii et al 2001a, 2001b)); for a throughprocess study, see (Jiang et al (2001)). These experiments contributed strongly to the insight
that the supercurrent in the tapes is not mainly limited by the pinning strength (Cai et al 1998)
but by the percolative flow around barriers on different length scales, such as grain boundaries,
secondary phases and cracks (Welp et al 1995a).
Figure 63 shows the flux distribution of a high-current (jc = 5.4 × 108 A m−2 at 77 K)
multifilamentary Bi-2223 tape in plan view. Magnetic flux penetrates not only along the axis
longitudinal to the tape in the intersections between the individual filaments. In addition, there
appears a periodic pattern of transverse flux penetration (length scale ≈0.5 mm) characteristic
748
Ch Jooss et al
Figure 64. From top to bottom: a light micrograph of the cross section of a silver-sheathed Bi-2223
multifilament tape. A series of SEM micrographs of three types of filament, I–III, each shown in
overview and in detail with larger magnification. (I) The least deformed filament located at the
edge of the tape. (II) The filament in the intermediate zone. (III) The most deformed filament
in the centre of the sample. Note that the Bi-2223 phase is imaged in dark grey (Schuster et al
1996c).
Figure 65. Flux distributions in the three types of filament in plan view with a perpendicular
magnetic field of µ0 Hex = 14 (I), 27 (II), and 82 mT (III) at T = 5 K. The black lines indicate the
filament edges. The filaments were extracted from the tape by grinding the flat surfaces of the tape.
This avoids influences of the underlying stacks on the flux distribution (Schuster et al 1996c).
of the cracks produced by the intermediate rolling step required to densify the tape after its
first heat treatment (Parrell et al 1996). Also on smaller length scales, current percolation due
to cracks and grain boundaries can be found. Cai et al (1998) found features characteristic
for local c-axis transport in individual Bi-2223 filaments which were extracted from a highjc multifilament tape, although the current was injected to flow along the a–b plane of the
filaments (see figures 63(a), (b)). This indicates the presence of current barriers at basalplane-faced grain boundaries which are very common in the Bi tapes. The crossing of such
GBs requires c-axis current flow on a submicrometre scale. The existence of such GBs is also
confirmed by x-ray rocking curves showing a c-axis distribution of typically 13◦ (FWHM) for
the best tapes.
Current distributions in high-Tc superconductors
749
Figure 66. (a)–(c) Plan-view magneto-optical images at 11 K and 40 mT for high-jc Bi2223 monofilamentary tapes with jc equal to 2.7 × 108 A m−2 (a), 3.3 × 108 A m−2 (b) and
3.4 × 108 A m−2 (c). Dark regions shield the magnetic flux very well and exhibit high local jc s.
The flux easily penetrates into the cracks along the longitudinal direction (the rolling direction of
the tape). Together with some transverse cracks, they form a crack network. (d) A backscattered
electron image of the transverse cross section of one tape. (e) Part of the image (d) showing a crack
(Jiang et al 1999).
The texture dependence of the critical current density within individual Bi-2223 filaments
was also shown via magneto-optics by Schuster et al (1996c) (see figures 64 and 65). The
rolling may cause a spatial variation of the grade of deformation over the cross section of
the tape and, therefore, the grain alignment within individual filaments depends on their
position in the cross section of the tape. This observation is also confirmed by the work
of Welp et al (1995b). Further magneto-optical studies of grain alignment and the influence of
columnar defects introduced by carbon nanotubes in Bi-2212 tapes were made by Huang et al
(1999). The silver-sheathed multifilamentary Bi-2223 tapes in the experiments of Schuster
et al (1996c) were prepared by the standard PIT technique via bundling, drawing and rolling by
the Vakuumschmelze GmbH with critical current densities at T = 77 K up to 2.6 × 108 A m−2
over the length of about 30 m.
Typical representatives (Eibl 1995) of different filaments with different microstructures
are shown in figure 64. The uppermost image in figure 64 shows a light micrograph of the cross
section of the 55-filament tape. The shapes of the filaments depend on their position in the cross
section of the tape. The outer filaments are roughly ellipsoidal whereas the filaments in the
centre look like flat and wide dumb-bells. Equally deformed filaments are located on roughly
elliptic lines as sketched in the diagram in figure 65. The horizontal line through the mid-point
of the tape is a symmetry axis. The strongly different deformations of the filaments lead to
three classes of filaments, I, II, and III, with different microstructures as sketched in figure 64.
The least deformed filaments (I) in the outer regions of the tape exhibit irregularly
disordered superconducting platelets and large secondary phases. The amount and size of
secondary phases decrease with increasing grade of deformation and the Bi-2223 platelets are
more and more oriented parallel to the tape plane; see filaments II and III. At this stage of
deformation a pronounced growth of ‘spikes’ penetrating the Ag matrix is observed. Filament
III finally shows a good texture and very few secondary phases. The amount of secondary
phase decreases during the thermomechanical treatment of the samples, mostly in the more
strongly deformed filaments.
750
Ch Jooss et al
a
f
HT2 Pressed
Jc 11 kA/cm 2
b
g
HT2 Rolled
Jc 11 kA/cm 2
c
h
HT3 Pressed
Jc 21 kA/cm 2
d
i
HT3 Rolled
Jc 4 kA/cm 2
e
j
Tape axis
HT1
Jc 3 kA/cm 2
(77K,0T)
1 mm
Figure 67. Plan-view magneto-optical images of differently processed Ag-sheathed Bi-2223
monocore tapes after ZFC to 13 K. Images (a)–(e) were recorded at µ0 Hex = 24 mT and images
(f )–(j) at µ0 Hex = 40 mT. Note that pressed samples have defects which run parallel to the original
rolling direction, whereas the defects run across the core width in rolled samples (Parrell et al 1996).
Figure 65 shows the flux distributions of the three types of filament on their flat surfaces
at T = 5 K for applied magnetic fields of µ0 Hex = 14 (I), 27 (II) and 82 mT (III). The
magnetic flux penetrates into filaments I and II preferentially along the boundaries between
badly coupled superconducting grains. Since the intragrain critical current exceeds that through
the boundary by far, several not fully penetrated grains are visible near the edges. In filament
III a sharp flux front is observed which is only slightly disturbed by some ‘flux fingers’, which
shows the much better coupling of the superconducting platelets. The critical current densities
along the filaments are estimated to be jcI = 8 × 107 A m−2 , jcII = 1.8 × 108 A m−2 and
jcIII = 8 × 108 A m−2 at 77 K. The poorer textures of the filaments I and II as compared to
filament III reduce the current capabilities of these filaments by factors of jcIII /jcII = 4 and
jcIII /jcI = 10. The ratios of the critical currents are nearly independent of temperature.
In Bi-2223 monocore tapes the grain alignment and crack structure also strongly determine
the percolative current flow on different length scales (Parrell et al 1996, Jiang et al 1999).
Current distributions in high-Tc superconductors
751
Figure 66 shows the longitudinal crack structure in a high-jc tape processed in the Applied
Superconductivity Center, University of Wisconsin–Madison (Jiang et al 1998, 1999). Even
if the cracks are approximately oriented parallel to the current flow, the inhomogeneous crack
distribution forms crack networks which represent barriers to the supercurrent. This indicates
that the performance of Bi-2223 tapes could be further improved by better controlling the
deformation conditions and the liquid phase to better avoid and to heal the cracks during
thermomechanical treatment.
The direction of the cracks and therefore the magnitude of the critical current density in
Bi-2223 monocore tapes can be directly influenced by the processing method. Parrell et al
(1996) made a comparative magneto-optical study of rolled and pressed Ag-sheathed tapes
with different heat treatments (see figure 67). Figures 67(a) and (f ) show the PIT sample
before mechanical deformation after one heat treatment of 50 h at 825 ◦ C in flowing 7.5%
O2 /N2 atmosphere in two different magnetic fields. The images show a granular flux pattern
corresponding to a low transport current density jc of 3 × 107 A m−2 at 77 K.
After pressing (figures 67(b) and (g)) or rolling (figures 67(c) and (h)), a second heat
treatment for 100 h is performed, increasing the transport jc at 77 K in both kinds of sample
to a value of 1.1 × 108 A m−2 . During the second heat treatment, the Bi-2212 content was
reduced from 30 to 5% and the Bi-2223 content increased to 95%. After a second deformation
and a third heat treatment the samples contain no Bi-2212 phase. The flux patterns after this
step are shown in figures 67(d) and (i) for the pressed (transport jc = 2.1 × 108 A m−2 ) and
in figures 67(e) and (j) for the rolled sample (transport jc = 4 × 107 A m−2 ). Note that all
images in figure 67 are done after the heat treatment.
The flux patterns in figure 67 clearly show the generation of preferentially oriented cracks
during the deformation process which are not fully healed during the heat treatment. Whereas
in the pressed samples the cracks are oriented parallel to the tape axis, the rolled samples exhibit
a transverse crack structure which disturbs more seriously the current flow compared to the
longitudinal cracks. These experiments also give insight into one of the central optimization
problems for the Bi-2223 tapes (Anderson et al 1999): the deformation process exerts a
beneficial effect on the grain alignment and the density of the core; however, it is responsible
for cracking, and a healing of the cracks during the heat treatment requires a sufficient amount
of liquid phase in the core. For the rolled sample, the second deformation step was of little
value because the more jc -detrimental transverse cracks (compared to the longitudinal cracks
produced by pressing) were not healed. An additional factor is that rolling is less effective
than pressing in densifying the core (Parrell et al 1994). This is consistent with the markedly
lower contrast in the magneto-optical image.
For practical applications, it is of vital importance to study how Bi conductors will perform when subjected to realistic operating conditions. One of the most important factors is the
effect of strain on the critical current density. Magneto-optical imaging revealed significant
results on strain-induced current-limiting defects and critical strain values (Polak et al 1997,
2000, Polyanskii et al 1998). Figure 68 shows a series of flux distributions of a good-quality
Bi-2212 monocore tape before bending (a) and after bending to a strain of b ≈ 0.1% (b) and
0.3% (c). Before bending the tape had a jc of 2 × 109 A m−2 at 4.2 K. The flux-admitting
defects lie largely at the edges of the core, thus minimizing their influence on the transport
current Ic . After only ≈0.1% bend strain, cracks start to penetrate the central region of the
core without, however, completely crossing the filament and seriously interrupting the current
flow. In contrast, a bend strain of 0.3% induces cracking across the whole core and seriously
interrupts the current flow. Polyanskii et al (1998) also showed that regions of pre-existing
cracks or other pre-existing defects are more sensitive to bending, and severe damage of the
tape occurs already for strains of ≈0.1%.
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Figure 68. Magneto-optical images of a good-quality Ag-sheathed Bi-2212 monocore tape. (a)
The flux distribution in the original state with zero strain, (b) after bending to a strain of b = 0.1%
and (c) to b = 0.3%. All images were taken after ZFC to 16 K and applying an external field of
40 mT. The width of the tape is W ≈ 2 mm (Polyanskii et al 1998).
An interruption of the current path in Ag-sheathed Bi tapes may be related to a current
shunting through the silver sheath. It has been shown that this effect can play an important
role in determining the transport properties of the composite conductors (Gurevich et al 1993,
Suenaga 1995, Welp et al 1995a). A distributed resistance model predicts that a characteristic
current-transfer length, λtr , governs the current flow around the barriers in the silver sheath
(Fang et al 1996a and 1996b). A magneto-optical study by Peterson et al (1997) yields an
experimental value of λtr of 0.56 mm at 77 K. This suggests that for a conductor with a crack
spacing closer than λtr , a current will flow in the silver sheath along the entire conductor length.
Figure 69 shows the temperature dependence of the current shunting of a Bi-2223
monocore tape containing a single crack across the core. At 30 K the applied transport current
of I = 21 A remains undercritical for the crack and all the current flows in the superconducting
core. At 110 K, all the current flows in the silver causing a strong flux contrast at the silver
edge, whereas at 90 K the current flows partially in the silver as indicated by the two d + -lines
near the crack. They indicate that the current flow lines are strongly bent around the crack and
a significant part of the current is flowing in the silver.
The presence of currents in the silver matrix together with the presence of local
superconducting bridges between neighbouring filaments may give rise to a significant coupling
of individual filaments in a multifilament conductor. Such a coupling leads to a substantial
increase in ac loss. A study of ac losses in a Bi-2223/Ag tape has shown that the filaments
behave as if they are connected into bundles with a typical number of eight filaments (Gömöry
et al 1998). Simultaneously, multifilamentary conductors benefit from filament coupling, since
the current flow can be maintained when it encounters a crack or barrier within a single filament.
A quantitative analysis of the temperature-dependent filament coupling in a 55-filament
Ag-sheathed Bi-2223 tape was done via magneto-optical imaging by Bobyl et al (2000); see
also Welp et al (1995a). Figure 70 shows the flux distribution (planar view TEM image) at
the surface of the tape. The peaks in the flux profiles of figure 70(d) correspond to the bright
stripes in the magneto-optical images. With increasing temperature the amount of trapped
flux in the individual filaments decreases as expected from the temperature dependence of the
Current distributions in high-Tc superconductors
753
y
x
y
(a)
x
OG 21 A 30 K
(b)
(c)
OG 21 A 90 K
OG 21 A 110 K
Figure 69. Top: the flux distribution of a Ag-sheathed Bi-2223 monocore tape with a crack after
ZFC to 30 K and application of µ0 Hex = 10.7 mT. The width of the frame corresponds to a length
of 3 mm. Bottom: the flux distribution at an edge of the tape due to a transport current of 21 A at
30 K (a), 90 K (b) and 110 K. The horizontal dashed lines mark the edges of the superconducting
core and of the silver, respectively. The grey arrows in (b) indicate the current path around the
crack. Brightness and darkness correspond to positive and negative flux (Peterson et al 1997).
vortex pinning force. Additionally, the flux profiles become smoother which can be seen most
clearly in the region −0.9 mm < x < 0. At low temperatures one finds three distinct peaks in
the B-profiles corresponding to the three filaments marked with arrows in figure 70(a). As the
temperature increases, the two minor peaks gradually vanish and finally disappear at 70 K. The
corresponding intrafilament and interfilament current densities in 70(e) were obtained by fitting
a strip model to the profiles. It is shown that the interfilament current density exhibits a weaker
temperature dependence and dominates over the intrafilament current density above 90 K.
Similar to the methods for joining of YBCO melt-textured bulk samples, ways of joining
Bi tapes have been developed, in order to fabricate long conductors (Tkaczyk et al 1993,
Lee et al 1995, Hase et al 1996). A magneto-optical study of the connection of different joints
in Bi-2223 tapes was made by Koblischka et al (1998b).
In summary, magneto-optical investigations of Bi tapes have contributed strongly to the
progress towards understanding the current-limiting mechanism and the percolative current
flow in these complex multicomponent conductors. Taking the results reviewed in section 7
as a whole, it can be concluded that the magneto-optical measurement technique is one of the
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Ch Jooss et al
Figure 70. Filament coupling of a 55-filament Bi-2223 tape. (a) An optical image of the tape cross
section; three filaments at the surface are marked with arrows. (b), (c) Trapped flux in the tape after
applying a large external field and decreasing it to zero at 19 (b) and 108 K (c). (d) Flux density
profiles in the remanent state at different temperatures obtained by averaging over 100 µm wide
bands across the tape. The bands are shown under the profiles. (e) The temperature dependence
of the intrafilament and interfilament current densities. The data were obtained by fitting current
distributions to the flux profiles in (d) (Bobyl et al 2000).
key tools for investigating the relation between microstructure and current-carrying capability
in not perfectly textured and non-textured materials. It is shown that the current pattern in
a superconductor is strongly determined by the grain structure and the grain boundaries of
the material. Using magneto-optics this correlation can be demonstrated very well and it is
clearly shown that a higher degree of texture can strongly improve the characteristics of the
superconductor. However, it is not only the local grain boundary properties that are important
for the magnitude and the pattern of the percolative current flow through complex materials.
Other properties, such as the length of the GBs, the ratio of the samples width to the grain size,
microcracks, precipitates, interfaces between different materials and the anisotropy are also
important. Numerical studies of the current flow through connected Bi filaments show that
under certain circumstances, the anisotropy of the current densities contributes more strongly
to the current suppression than the GB itself (Zeimetz et al 2000). As regards further materials
optimization for applications of HTS, magneto-optical imaging will definitely continue to play
an important role in the future.
Current distributions in high-Tc superconductors
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8. Pinning and depinning mechanisms
Flux and current distributions of HTS on mesoscopic and macroscopic length scales are
all based on the microscopic interaction of vortices with material defects. Most features
of the observed flux and current patterns can be understood in the framework of the Bean
model as a result of static equilibrium between vortex pinning and driving forces. A more
developed description is given by Brandt’s non-linear conductor model which allows one also
to include time-dependent and electric field-dependent processes like sweeping the external
field or current, flux creep and flux flow. Pinning causes a strongly non-linear current–voltage
characteristic of the superconductor which is described by essentially two parameters, the
critical current density jc at which the Abrikosov vortices depin at T = 0 and start to move
under the action of the Lorentz force and the activation energy U . The resulting flux penetration
on a mesoscopic level is well described theoretically by model calculations using a current–
electric field law, e.g., of the form E ∼ (j/jc )n in all regimes from thermally activated flux
flow (n = 1) to the classical Bean model (n 1), where E is the electric field and the exponent
n is determined by U (j ) (see section 4).
In addition to the visualization of current patterns, magneto-optical imaging is also a
powerful tool for the determination of microscopic pinning and depinning mechanisms. In
particular, this is the case if the underlying microscopic mechanisms induce modifications of
the mesoscopic current patterns, such as an anisotropic jc or a spatial variation of jc . It is
far beyond the scope of this article to review pinning in HTS (see e.g. the reviews of Brandt
(1995a) and Blatter et al (1994)). We give here only three examples of magneto-optical studies
of pinning and depinning mechanisms at columnar defects (section 8.1), antiphase boundaries
(section 8.2) and surface pinning (section 8.3) which have been selected using the following
criteria: (i) the defect structure in the crystal lattice is very well defined and characterized
which is necessary for a clear microscopic analysis of the results; (ii) a flux imaging method
has to be applied because the particular mechanism can be identified most clearly by measuring
the spatial distribution of the current density.
8.1. Columnar defects
√
Linear defects (LDs) with a radius of the non-superconducting core ξv ≈ 2ξ (ξ is the
coherence length) are very effective pinning centres (Brandt 1980). In HTS such defects
can be introduced by high-energy heavy-ion irradiation (Bourgault et al 1989, Hensel et al
1990, Watanabe et al 1991, Hardy et al 1991b, 1992a, Chenevier et al 1992). Various groups
(Roas et al 1990, Civale et al 1991, Konczykowski et al 1991, Hardy et al 1991a, 1992b,
Gerhäuser et al 1992, Leghissa et al 1992, Schuster et al 1992b, Budhani et al 1992) report
an irradiation-induced shift of the irreversibility line (IL) to higher temperatures T and higher
magnetic flux densities B and an enhancement of jc . The jc -enhancement can be attributed to
strong, correlated pinning of individual vortices trapped over a large length to the LDs. The
pinning mechanism in this case is the spatial variation of the vortex condensation energy in the
normal-conducting LDs, the so-called ‘δTc effect’ (for other pinning mechanisms see section
8.2). If the length scale of the order parameter suppression fits approximately the diameter of
the vortex core, a maximum pinning force per unit length fp ≈√0 /ξ is reached (0 is the vortex
line energy per unit length in the bulk superconductor). For 2ξ R λ the variation of
the kinetic energy of the vortex eddy currents may contribute to the pinning.
In contrast to the relatively simple behaviour of the single-vortex pinning force, a great
variety of different behaviours appear on the level of collective properties of vortices in an
anisotropic superconductor with statistically distributed LDs. Using the Träuble–Essmann
decoration technique (Essmann and Träuble 1967, 1971) to visualize the FLL, Leghissa et al
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(1993) and Dai et al (1994) found a highly disordered vortex state due to randomly distributed,
very effective linear pins in Bi-2212 single crystals. In particular, different modes of vortex
motion and depinning mechanisms based on the creation and motion of pancake vortices,
vortex kinks, double kinks and loops have to be considered. In Bi-2212 single crystals and
Bi-2223 tapes at low temperatures, the small enhancement of U by the introduction of LDs was
attributed to the two-dimensional pancake-like structure of the vortices (Gerhäuser et al 1992,
Leghissa et al 1992, Kummeth et al 1994). For these pancake vortices (Clem 1991) (see also
(Bulaevskii 1973)), U (j ) remains low because pancake segments can depin individually from
the LDs. This is in contrast to the case for the three-dimensional (3D) HTS DyBa2 Cu3 O7−δ
(DBCO) (Schuster et al 1992b) and YBCO (Konczykowski et al 1993, Klein et al 1993a),
where it was suggested that the larger enhancement of jc and U (j ) is limited by the low energy
of vortex–kink formation between LDs (Nelson and Vinokur 1992). Theoretical treatments of
FL pinning in samples with LDs using a Bose-glass model and their application to experimental findings are given in Nelson and Vinokur (1993); see also Lyuksyutov (1992), Hwa et al
(1993a), Jiang et al (1994), Budhani et al (1994), Moshchalkov et al (1994).
8.1.1. Inclined parallel columnar defects. Deeper insight into vortex pinning and depinning
is obtained in experiments, where the FLs are aligned parallel or inclined to the LDs: in HTS
with 3D vortices (e.g. YBCO), the jc -enhancement is large when the applied magnetic field Hex
is aligned with the LDs, whereas only a small enhancement is observed when the LDs deviate
from the magnetic field direction (Civale et al 1991, Kraus et al 1994); for YBCO thin films
see Holzapfel et al (1993) and Prozorov et al (1994). This is in contrast to the case for Bi-2212
with 2D vortices, where no angular dependence was found for T < 50 K (Klein et al 1993b,
Thompson et al 1992). At higher temperatures, the difference between the jc -enhancements
for the different directions of Hex parallel and inclined to the LDs reappears and is attributed to
a crossover from 2D to 3D vortices in such samples near T = 50 K. Such a 2D–3D crossover
was also observed in Bi-2223 tapes by Kummeth et al (1994). In YBCO/PrBa2 Cu3 O7−δ
(PBCO) superlattices the absence of an angular dependence was explained by the decoupling
into pancake vortices in the superlattices (Holzapfel et al 1993).
Vortices inclined to LDs cannot be described in the framework of the Bose-glass model
and an extended theory is required (Hwa et al 1993b). In infinitely extended samples with
vortices parallel to the LDs, depinning can occur only with creation of kink pairs in the volume,
as considered by Brandt (1992b, 1992c) and in the Bose-glass model which represents a
statistical theory of thermal depinning from randomly distributed LDs (Nelson and Vinokur
1992, 1993). However, in real experimental situations the flux lines are usually curved due to
the demagnetization effects at finite samples and consequently are inclined to the LDs. In this
case, the vortices have to form kinks from one LD to another which can slide easily along the
LDs. This effect should be strongly pronounced in samples with LDs inclined to the sample’s
surface as depicted in figure 71(a).
In HTS with 3D vortices and inclined LDs one expects an anisotropy of the critical current
densities flowing in directions different to the inclination direction (see figure 71(a)): for j
flowing parallel to the inclination direction, vortex depinning can only take place by double
kink formation (process 1) as predicted in the Bose-glass model. In contrast, the depinning
and motion of vortices driven by j⊥ can take place by the sliding of kinks along the LDs
(process 2), being impeded only by the background pinning which is present already before
irradiation and j⊥ should not be affected by introduction of LDs. Thus, for oblique irradiation
one expects a huge anisotropy ratio j /j⊥ ≈ 30 at T = 60 K (this is the ratio between the
critical current densities in irradiated and unirradiated crystals), due to the different depinning
mechanisms 1 and 2 in figure 71(a) (Schuster et al 1995c).
Current distributions in high-Tc superconductors
757
Figure 71. (a) A sketch of three different modes of vortex motion parallel and perpendicular to
the inclination direction of the LDs. Process 1: vortex motion perpendicular to the LDs forced by
j . It is possible only after thermal activation and spreading of kink pairs in the sample volume.
Process 2: vortex motion in the inclination direction of the LDs by kink sliding forced by j⊥ . For
a continuous motion, kink nucleation at the surface is needed. Process 3: perpendicular motion as
for process 1 but due to kink nucleation at the surface which is easier than kink pair nucleation. (b)
Kink nucleation at the surface for vortex motion parallel (process 2) and perpendicular (process
3) to the LDs. The nucleation energy is largest for nucleation mode C and smallest for nucleation
mode A. (c) Current distribution and d-lines for a sample with an anisotropy j /j⊥ due to inclined
LDs and a difference in j⊥ (see figure 72).
Figure 72 shows the magnetic flux distribution of a DBCO single crystal before
(figures 72 (b), (c)) and after (figures 72 (e)–(g)) inclined heavy-ion irradiation and the
corresponding current distribution in the critical state (Schuster et al 1995c). Before irradiation,
the single crystal shows an isotropic flux penetration of a typical rectangular sample which
is slightly disturbed by the presence of some twin boundaries. The flux distributions after
irradiation are visualized at T = 60 K in figures 72(e)–(g). The ion beam was directed
parallel to the longer crystal edge and inclined to its surface by ϕ = 45◦ . The black frames
mark the crystal edges. The flux penetration differs considerably from that observed in the
unirradiated crystal: from the anisotropic penetration depths of the flux front s and s⊥ (see
also figure 71(c)), one finds an anisotropic current distribution, where the j flowing parallel
to the inclination direction of the LDs is larger than j⊥ .
From figure 72(g) a ratio of j /j⊥ ≈ 2 is obtained which is reproduced also in other
experiments with 45◦ -inclined 0.9 GeV Pb-ion irradiation (Schuster et al 1994b). Obviously,
this anisotropy ratio is much smaller than expected from the considerations above, where
j /j⊥ ≈ 30 is predicted from the different activation energies of kink pair nucleation (j ) and
kink sliding (j⊥ ).
As was pointed out by Schuster et al (1995c), depinning processes are not only induced
by vortex kink pairs in the volume but are also produced by single kinks created at the sample
surface. If one assumes that the energy for nucleation of a kink at the surface is only half
of that for a kink pair, it follows that the surface kink nucleation plays the main role for flux
motion. As depicted in figure 71(b), three different modes of kink nucleation at the sample
surface can be considered:
(A) Kink nucleation across the acute angle between surface and LD. This nucleation energy
is lowest because the vortex can be shortened by forming a kink.
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Ch Jooss et al
i
Figure 72. Flux distributions in a DBCO single crystal before and after irradiation with 500 MeV
Xe ions at ϕ = 45◦ and φt = 1.29 × 1011 ions cm−2 . The observation temperature is T = 30 K
before irradiation ((b), (c)) and T = 60 K after irradiation ((e)–(g)). (a) A polarization micrograph
of the crystal surface with the twin structure. (b) µ0 Hex = 171 mT, (c) 256 mT. (d) Isotropic
current distribution in a rectangular sample in the critical state. (e) µ0 Hex = 85 mT, (f ) 171 mT,
(g) 256 mT. (h) Anisotropic current distribution in a rectangular sample. MOLs: EuSe (before
irradiation) and an iron garnet indicator (after irradiation). (i) A sketch of the angular irradiation
introducing parallel LDs which are inclined to the sample surface (Schuster et al 1995c).
(B) Kink nucleation perpendicular to the inclination direction. The vortex now has to be
elongated to reach the neighbouring defect and thus the nucleation energy is larger than
for mode A.
(C) Kink nucleation at the obtuse angle between the surface and LD in the sliding direction.
In this case the nucleation energy is largest and therefore mode C does not play a role in
the vortex motion.
The anisotropy j /j⊥ is thus expected to be due to the different nucleation energies for
kink generation modes A and B which are responsible for the vortex motions parallel and perpendicular to the inclination direction, respectively. As illustrated in figure 71(b), for vortex
motion from left to right, the kinks will nucleate by mode A at the lower surface 1, whereas
for the opposite motion the kinks nucleate by mode A at the upper surface 2. Thus, different
surface qualities or different diameters of the LDs at the two surfaces should cause different
potentials for kink nucleation and induce an additional difference in j⊥ for flux penetration
from opposite sides. Such a difference is indeed seen in figure 72(g) where the double-Y structure is not exactly centred but shifted towards the side where j⊥ is larger. From figure 72(g)
one obtains j /j⊥(1) = 2.0 for kink nucleation at surface 1 and j /j⊥(2) = 2.8 for kink nucleation
at surface 2. For this case the current distribution is sketched in figure 71(c).
A direct proof that jc is limited by depinning processes which are induced by surface kink
nucleation was obtained recently by Indenbom et al (2000) by investigating the magnetic flux
Current distributions in high-Tc superconductors
759
Figure 73. (a) A polarized light photograph of the surface of a YBCO single crystal with a
large surface step (indicated by arrows) dividing into 10 and 20 µm thick parts. (b) and (c)
show the remanent induction on the crystal surface after the application and removal of a field
µ0 Hex = 36 mT parallel to c and parallel to the ion tracks: (b) before irradiation, T = 40 K;
(c) after irradiation with 6 GeV Pb ions, T = 80 K. The arrows indicate the position of the step.
(d) Surface depinning of a vortex (bold line) from columnar defects (cylinders). Short arrows
indicate the vortex kink sliding down from the surface, creating a shift of the vortex. On the right
side the surface critical current is sketched (Indenbom et al 2000).
penetration into a 6 GeV Pb-irradiated YBCO single crystal with a surface step. The irradiation
was performed with tracks parallel to the c-axis with a density of 5 × 1010 cm−2 . Figure 73
shows the flux penetration into the YBCO crystal before and after the irradiation. Supposing
that one has bulk pinning and thus a homogeneous jc in the full thickness d of the crystal, the
penetration depth of the flux is proportional to the sheet current Jc = jc d. This is visible in
figure 73(b) for the unirradiated sample where flux penetration into the right-hand side of the
sample is much easier than into the thick left side of the crystal.
After irradiation the flux pattern is drastically changed. Because of a strong increase in
the critical current, the temperature for the flux distribution in figure 73(c) had to be increased
in order to observe a penetration depth which is comparable to that before irradiation. Now
the flux penetrates equally far into the thick and the thin parts of the sample, which directly
shows that surface depinning processes via kinks determine jc in the sample.
8.1.2. Crossed columnar defects. From the experiments shown above it follows that it
would be desirable to suppress kink sliding to achieve a further improvement of jc and the
irreversibility line. Using non-parallel LDs it was possible to increase jc and the irreversibility
temperature Tirr above that of parallel LDs (Hwa et al 1993a, Civale et al 1994, Krusin-Elbaum
et al 1994). Kink sliding should now be stopped by trapping of the vortex kinks at the LDs
with different orientations.
Figure 74 shows the flux distribution in a cross-irradiated DBCO single crystal (0.9 GeV
Pb ions with an irradiation angle of ϕ = ±45◦ ) for a fully penetrated state at different
temperatures (Schuster et al 1995c). The white frame marks the sample edges. After
irradiation, an anisotropic current distribution is observed. The anisotropy ratio j /j⊥ increases
with temperature from a value of 1.1 (20 K) to 2.3 (80 K). The change of j /j⊥ is attributed to
different T -dependences of j and j⊥ . The curve j /j⊥ (T ) is plotted in figure 74(k).
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Figure 74. Flux distributions in the critical states in a cross-irradiated DBCO single crystal at
various temperatures. (a) T = 30 K, µ0 Hex = 363 mT, (b) 40 K, 288 mT, (c) 50 K, 235 mT, (d)
55 K, 213 mT, (e) 60 K, 181 mT, (f ) 65 K, 160 mT, (g) 70 K, 128 mT, and (h) 75 K, 96 mT. (i), (j)
Sketches of vortex motion in the presence of crossed LDs: (i) kinks of vortices 1 and 2 which are
pinned by crossed LDs slide along the LDs; (j) the kinks of vortices 1 and 2 are pinned by the
next intersection of the LDs. The vortex bend of vortex 2 at the LD intersection is more strongly
pinned than that of vortex 1 which can be depinned in its sliding direction. The depinning mode
B—by the usual kink pair nucleation—is equal in all directions perpendicular to c and requires an
intermediate energy. (k) The temperature dependence of the anisotropy ratio j /j⊥ of the sample
in (a)–(h). The j /j⊥ ratio strongly increases with T (Schuster et al 1995c).
The above data are explained by figures 74(i)–(j). Sliding kinks are trapped by
intersections of LDs and therefore the migration of surface kinks as a relevant depinning
process is suppressed. This follows also from the observation that the additional anisotropy of
j⊥ for oppositely penetrating flux, which was attributed to surface effects, has disappeared
(compare figures 71(c) and 74). If kink sliding is stopped, the vortices are pinned over
their full length by the LDs and depinning now proceeds only by kink pair formation, which
results in a much higher activation barrier. Thus, the activation energy of vortices pinned
by crossed LDs becomes similar to the considerations in the Bose-glass model (Nelson and
Vinokur 1992, 1993) and the irradiation-enhanced critical currents are considerably larger
than for parallel irradiation at the same fluence. The observed anisotropy ratio j /j⊥ > 1 is
explained by different activation energies for bent and straight flux lines (see figure 74(j)).
For depinning mode C, the vortex has to be considerably more curved and stretched. Since
this requires the highest activation energy it is therefore not relevant for flux motion. For
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761
understanding the anisotropy of j /j⊥ , it is necessary to compare depinning mode A (where
the vortex is straightened and yields a lower activation energy) with depinning mode B—by
kink pair nucleation. The latter is equivalent for all directions of vortex motion and requires
an intermediate activation energy (compared to A and C). Vortices which are driven by j and
move perpendicular to the inclination direction of the crossed LDs can depin only by mode B
with a higher activation energy. This explains why j > j⊥ and j /j⊥ increases with increasing
T in figure 74(j).
Similarly to in the experiments on DBCO, in Bi-2212 single crystals the jc -enhancement
due to crossed irradiation is much higher than for parallel irradiation. However, in Bi-2212 no
anisotropy of the depinning modes is observed (Schuster et al 1995c). This can be explained
by the pancake structure of the vortices in Bi-2212, where the motion of vortices is initiated
by one pancake and different activation energies of differently bent 3D vortices play no role.
8.2. Pinning mechanism at antiphase boundaries in thin films
The largest jc ≈ (2–5) × 1011 A m−2 at T = 4.2 K (jc ≈ (2–6) × 1010 A m−2 at T = 77 K) in
HTS are usually observed in highly textured epitaxial YBa2 Cu3 O7−δ films on single-crystalline
substrates. They exhibit an intrinsically complex and rich microstructure which is created by
different growth modes such as island growth, Stranski–Krastanov growth or step flow growth,
depending on the substrate and the growth conditions. Due to the high density of different
lattice defects, there is a long-standing controversy over which of these defects are the most
effective as pinning sites responsible for the large jc . It was shown that point defects such as
oxygen vacancies (Theuss and Kronmüller 1991, Theuss 1993) are, in particular, important
pinning sites in HTS single crystals; however, their volume pinning force after statistical
summation seems to be too small (their jc remains three orders of magnitude smaller than in
epitaxial thin films) to account for the large jc in epitaxial films (for an opposite statement
see Griessen et al (1994)). Small precipitates (Eibl and Roas 1990, Yasuda et al 2000) (for
theory see also Matsushita (2000)) can significantly improve the jc in single-crystalline and
melt-textured materials (see section 7.3), but the jc is still two orders of magnitude smaller
compared to that for epitaxial thin films. Since in thin films surface pinning effects could be
important, growth steps and growth terraces (McElfresh et al 1992, Schalk et al 1996) and
the role of surface roughnesses for flux line pinning (Jooss et al 1996a, 1996b) have been
considered. Depending on the roughness and film thickness, this effect, however, can account
for ≈10–30% of the thin-film jc , only. We come back to this issue in some detail in section 8.3.
In general, the large jc in thin films requires vortex pinning for correlated disorder in the
crystal lattice, such as columnar and planar defects. This allows a proper alignment of the
vortex system with the defect structure, resulting in higher pinning forces compared to the case
for collective pinning at pointlike defects (Blatter et al 1994). Consequently, one-dimensional
correlated disorder generated by e.g. screw dislocations (Mannhart et al 1992), edge dislocations (Dam et al 1999) and dislocation chains (Schalk et al 1996) has been attributed to
strong pinning of vortices and is able to explain the large observed critical current densities
for moderate flux densities.
Since their anisotropic pinning force is ideally suited for space-resolved investigations
by magneto-optical flux imaging, we focus in this section on the two-dimensional correlated
disorder of planar defects which can also account for large jc (Zandbergen et al 1990b, Pan
1993, Misra et al 1994). As regards island growth-related planar defects, highly textured
epitaxial YBCO films with large jc can be imaged as a network of planar defects, consisting
of LAGBs, twin boundaries (TBs) and antiphase boundaries (APBs). For the understanding
of the pinning properties of two-dimensional correlated disorder, an in-depth investigation of
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the transverse critical current jc,T as well as jc,L longitudinal to current-carrying boundaries is
essential. In the following, the directions which are transverse and longitudinal to boundaries
(both perpendicular to the c-axis) will be called L and T , respectively. Planar defects exhibit
a complex and sometimes contradictory behaviour with respect to the critical state properties.
TBs in single crystals (Belyaeva et al 1992, 1993, Duran et al 1992, Turchinskaya et al 1993,
Zhu and Welch 1999) represent strong pinning sites for vortices perpendicular to the boundary
(Vlasko-Vlasov et al 1994a, Li et al 1993), i.e. they lead to high jc,L . On the other hand, TBs
may give rise to vortex channelling along the boundary plane (Vlasko-Vlasov et al 1994a,
Li et al 1993, Wijngaarden et al 1997, Oussena et al 1995) and reduce jc,T (for theroy see
Shklovskii et al 1999). A similar behaviour was observed at APBs in thin films (Haage et al
1996, 1997). The longitudinal critical current density jc,L in arrays of parallel APBs exhibits
extraordinarily large values up to 30% of the depairing current density j0 ≈ 3 × 1012 A m−2 .
Remarkably, at the same time, the transverse jc,T crossing the boundaries reaches values of
jc,T ≈ 1–3 × 1011 A m−2 (4.2 K) which is on the same order as observed in highly textured
YBCO epitaxial films on well oriented single-crystalline substrates (Jooss et al 1999). LAGB
represent the planar defects consisting of arrays of edge or screw dislocations. Diáz et al
(1998b) report on vortex pinning at edge dislocations of low-angle [001] tilt boundaries. We
summarize in this section magneto-optical experiments on YBCO films with arrays of parallel
planar APBs, where the microstructure is very well defined and the induced anisotropy of jc
allows a clear analysis of the pinning properties.
8.2.1. Arrays of antiphase boundaries. For the investigation of vortex pinning at planar
defects of different structural widths, two kinds of YBCO film were used, both with a high
density of parallel antiphase boundaries. YBCO films of the first kind are grown on almost
perfect and stable SrTiO3 (106) surfaces and exhibit a very regular array of parallel APBs with
a small structural width of rp ≈ 0.4–0.7 nm and a mean distance of 6–8 nm (see figure 75).
Films of the second kind are grown on SrTiO3 surfaces which deviate slightly from the (106)
surface. In addition to the APBs, such films exhibit a larger density of extended APBs (EDs)
with rp ≈ 2 nm. This anisotropic microstructure is well reflected by the mean free path
of the charge carriers. Whereas the mean free path in the transverse direction lT ≈ 2–
6 nm corresponds well to the mean distance between planar APBs, the mean free path in the
longitudinal direction lL ≈ 70 nm reaches the values of YBCO single crystals. With this very
well defined microstructure, YBCO films on SrTiO3 (106) represent an excellent model system
for the investigation of vortex pinning and pinning mechanism at planar defects. For details
see the articles of Haage et al (1997) and Jooss et al (2000a).
Figure 76 shows a quantitative analysis of the current density distribution of a 55 nm thick
YBCO film grown on perfect SrTiO3 (106). The magneto-optically observed magnetic flux
density distribution is depicted as a greyscale image together with the current streamlines and
two profiles of the current density in the L-direction and the T -direction, respectively. Due
to the anisotropic pinning force of the planar APBs, the flux penetrates more easily in the
L-direction than in the T -direction. The critical current density exhibits exceptionally large
values up to jc,L = 8.5 × 1011 A m−2 in the L-direction and jc,T = 2–3 × 1011 A m−2 in the
T -direction, although jc,T has to cross the defect planes of the APBs.
If the misorientation of the substrate surface deviates slightly (≈±0.5◦ ) from a nearly
perfect SrTiO3 (106), jc,T is reduced to magnitudes below 1011 A m−2 and also jc,L is depressed
to ordinary values of 1–2×1011 A m−2 (Jooss et al 2000a, 2000b). This is due to a higher density
of macro-steps (MS) and kinks at the terrace edges of the substrate surface, forming EDs with
an extended distorted region of several nm. Extraordinarily large jc,L -values as well as nice
jc,T -values are reached only for YBCO films on almost perfect (106) surfaces without EDs.
Current distributions in high-Tc superconductors
763
a)
b)
Figure 75. Microstructure of YBCO films on SrTiO3 (106): (a) an STM image of a
100 nm × 100 nm area of a SrTiO3 (106) substrate after UHV annealing; (b) a schematic diagram
of the observed microstructure of YBCO films on SrTiO3 (106) substrates as revealed by TEM.
Due to partial overgrowth, the mean distance between APBs perpendicular to the defect plane is,
at 6–8 nm, somewhat larger than the terrace width of 2.3 nm on the SrTiO3 substrate. The APBs
are extended along the c-axis over some unit cells of YBCO. They are terminated by the inclusion
of a stacking fault (SF) in the (001) plane simultaneously creating a new APB in a region which is
shifted some nm in transverse direction.
D
S
D
F
y, T
F
x, L
Figure 76. The magneto-optically observed flux density distribution in a square-shaped, 55 nm
thick YBCO film on SrTiO3 (106) at µ0 Hex = 24 mT and T = 5 K. Full lines are indicating the
current path; current density profiles are plotted as the dashed lines. Differences from the case
for a homogeneous YBCO square with anisotropic jc (see for comparison figure 37) are due to
a macroscopic scratch (S), growth distortions (D) and other distortions whose size is some µm
(F). (S) divides the square sample into two parts which are only weakly connected by the flowing
supercurrents. The large defects (F) in regions where the flux penetrates in the L-direction create
large flux filaments.
8.2.2. Longitudinal pinning. For an understanding of the observed jc,T , it is not sufficient to
consider how a planar defect suppresses the depairing current density j0 on microscopic length
scales. Since magnetic flux lines are present in the sample, a finite jc,T -value requires that
764
Ch Jooss et al
b)
a)
y, L
x, T
0,25 mm
Figure 77. (a) The flux density distribution of a square-shaped, 170 nm thick YBCO film on 9.5◦
SrTiO3 at µ0 Hex = 66.4 mT and T = 5 K observed by means of EuSe as the magneto-optical layer
(spatial resolution 1 µm). This YBCO film exhibits a higher density of EDs and the filamentary
flux pattern is more pronounced as compared to figure 76. No filaments are visible in the region of
T -penetration of the flux. (b) A sketch of the longitudinal flux penetration in a YBCO film with
parallel APBs and EDs. The density of antiphase boundaries (6 nm) is much larger than indicated
in the drawing. The white area is filled by magnetic flux whereas the dark area is still flux free.
Ellipses represent Abrikosov–Josephson vortices with different anisotropies.
they are pinned by material defects parallel to the boundary. An analysis of this longitudinal
pinning force will be given in the following.
In general, the magnetic flux penetrating along the L-direction exhibits a filamentary
flux pattern extended macroscopically from the film edge towards the sample’s centre. They
indicate planes of lowered pinning strength with a mean distance in the T -direction of
x ≈ 60 µm which is much larger than the distance of the APBs and can be attributed
to the EDs. The filaments are strongly pronounced in samples with a higher numbers of EDs
(figure 77). Some apparent filaments in figure 76 marked with (F) are not related to EDs
but created by the d + -lines of macroscopic defects. In the case of an anisotropic jc , these
d + -lines degenerate to a couple of narrow, almost parallel lines which look similar to the
filamentary flux pattern at planar weak links, but, however, are not extended over the entire
sample along L.
The flux filaments, the longitudinal pinning mechanism and the decrease of jc,T with
increasing structural width of the planar defects are nicely described by applying a model of
Gurevich and Cooley (1994). At planar defects where the microscopic eddy current of vortices
is reduced from the (bulk) depairing current density j0 to the Josephson current density jJ,0 ,
the Abrikosov vortices turn into anisotropic Abrikosov vortices with a highly anisotropic
Josephson core. The vortex core of such AJ vortices is elongated along the planar defect with
a core radius of
√
3 3 j0
ξab ,
(68)
ξL =
4 jJ,0
√
whereas the transverse core radius remains at a size ξT ≈ 2ξab .
The longitudinal pinning force per unit length fp,L acting on AJ vortices would vanish for
a perfectly homogeneous planar defect. A non-vanishing fp,L results from inhomogeneities
of jJ,0 , e.g. due to variations in the structural width rp of the APBs or in their extensions along
the c-axis. The model of Gurevich and Cooley (1994) based on the variation of the Josephson
Current distributions in high-Tc superconductors
765
coupling energy of AJ vortices as a function of position within a planar defect gives
L
,
(69)
fp,L = δJ 0 2
L + ξL2
with inhomogeneity length L and disorder parameter δJ = δjJ,0 /jJ,0 . The line energy of
the vortex line is given by 0 = 20 /(4π µ0 λ2ab ). The longitudinal pinning force has its
maximum when L is of the order of the longitudinal core size ξL . Assuming a distribution of
inhomogeneity lengths L and AJ vortices that always take a pinning site with maximum fp,L ,
one may set L ≈ ξL and equation (69) leads to
1
fp,L ≈ δJ 0
.
(70)
2ξL
This means that the longitudinal pinning force decreases with increasing anisotropy of the AJ
vortices. Since the Josephson current of a ED with large rp is smaller than that of an APB,
according to equation (68) the AJ vortices at EDs exhibit larger anisotropies and therefore
fp,L for EDs is smaller. Within the framework of this simple model, using the experimentally
observed jc,T -values and assuming δJ ≈ 0.5, one obtains for the longitudinal core radius of
the AJ vortices ξL ≈ 5–7 nm at APBs and ξL ≈ 15–25 nm at EDs.
The presence of planar APBs and EDs with different longitudinal pinning strength is also
reflected by the magnetic field dependence of the transverse critical current jc,T . According
to Gurevich and Cooley (1994), the weakly pinned AJ vortices in the EDs are stabilized by
the more strongly pinned vortices in their neighbourhood (e.g. APBs with higher longitudinal
pinning forces), which gives
jc,T = fp,L (ED)/0 + c Bz .
(71)
Thus, jc,T increases with increasing local flux density approximately up to a value which is
given by the stronger pinning of the APBs. This is nicely seen in figure 78(a). For more details
and an analysis of the field dependence of jc,L (B), see Jooss et al (2000a); a similar behaviour
is found at LAGBs (Albrecht et al 2001). For the electric field distribution at APBs see Jooss
and Albrecht (2002).
8.2.3. Transverse pinning. In this section, the possible transverse pinning mechanisms
of vortices at planar defects are summarized theoretically and the theory is applied to an
analysis of APBs. In particular this is done to understand the extraordinarily large jc,L up to
8.5 × 1011 A m−2 at 4.2 K and the observed angular dependence of jc . Transverse pinning at
planar defects may occur by magnetic interaction, where the deformation of the eddy currents
of a vortex as a function of the distance to the defect plane gives rise to a pinning potential.
According to Gurevich and Cooley (1994) the transverse pinning force per unit length of a
vortex at a distance ξab < x ξL is given by
1
(72)
fp,T (x) = − 0 ,
x
where 0 = 20 /(4πµ0 λ2ab ) is the self-energy of the vortex line per unit length.
For core pinning, microscopic theory (Thuneberg 1989) suggests two fundamentally
different mechanisms for the suppression of the pairing amplitude in HTS: a direct suppression
of superconductivity by local variations in the coupling constant (e.g. due to hole depletion)
which gives rise to fluctuations in Tc (δTc pinning) and a suppression of the pairing amplitude
due to quasiparticle scattering (δL pinning). For the δTc effect, the transverse pinning force
per unit length of a planar defect with a reduced Tc can be estimated as (Blatter et al 1994)
0 Lp δTc
fp,T ≈
,
(73)
ξab d Tc
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Ch Jooss et al
Figure 78. (a) The dependence of jc,T and jc,L on the local magnet flux density Bz at different
external fields µ0 Hex = 32, 40, 48, 56.8, 64 and 80 mT at T = 5 K determined by magnetooptical measurements and inversion of the Biot–Savart law. The theoretical curve is calculated
from equation (71) with jT ,ED = 5.6 × 1010 A m−2 and c = 1.6 × 1011 A m−2 T−1/2 . (b)
The resulting change of the anisotropy ratio Aj of the critical currents as a function of the applied
external field Hex at 5 K.
where ξab the Ginzburg–Landau coherence length, Lp /d the pinned length of a vortex
normalized to the film thickness and δTc is the Tc -variation at the planar defect.
For the δl effect in superconductors with anisotropic gaps (e.g. d waves (Tsuei et al 1994,
Tsuei and Kirtley 2000)), the leading term of the scattering mechanism results also in a direct
suppression of the pairing amplitude (Thuneberg 1989, Franz et al 1996b, Friesen and Muzikar
1997, Jooss et al 1999). For this mechanism one can estimate a maximum transverse pinning
force per unit length (Jooss 1998) of
Lp
(74)
ptr,T j0 0 ,
d
where ptr,T denotes the transverse component of the transport scattering probability tensor.
Neglecting diffuse scattering, ptr,T = 1 − τ is determined by the transmission probability of
the plane τ . Whereas in isotropic s-wave superconductors, non-magnetic scattering affects
only the kinetic energy of vortex eddy currents (Anderson 1959, Zerweck 1981, Welch 1984,
Thuneberg 1982, 1989), in HTS, elastic scattering results in a change of condensation energy
due to the presence of an anisotropic gap. Recent work shows that the condensate is very
sensitive to non-magnetic scattering and the superconducting charge density ns is significantly
reduced by QP scattering whereas Tc remains more stable (Franz et al 1996a, 1997, Williams
et al 1998, Hirschfeld and Goldfeld 1993).
All transverse pinning mechanisms described above may lead to transverse pinning forces
per unit length with an order of magnitude of fp,T ≈ 10−3 N m−1 and, thus, to longitudinal
critical current densities of the order of 1012 A m−2 at T = 4.2 K, assuming δTc /Tc ≈ 1,
ptr,T ≈ 1 and Lp /d ≈ 0.5. Consequently, different pinning mechanisms at APBs cannot
be ruled out by considering the magnitude of jc,L alone. Additional studies of the angular
dependence of jc and studies of jc,L and jc,T as functions of the defect width rp are necessary
(Jooss et al 2000a).
In the following, a new magneto-optical method for measuring the angular dependence
of jc (αj ) (αj denoting the deviation angle of j from the T -direction) will be summarized. By
choosing appropriate sample geometries (e.g. discs or squares, where the edges are inclined
to the APBs), it is possible to force the current to flow with different αj with respect to the
APBs (see figure 79). The inversion of the Biot–Savart law allows one to image the current
fp,T ≈ 0.85
Current distributions in high-Tc superconductors
767
density components jT (x, y) and jL (x, y), respectively, for any part of the sample. The current
direction at any position (x, y) is then obtained from αj (x, y) = arctan(jL (x, y)/jT (x, y)).
Afterwards, the functions jT (αj ), jL (αj ) and jc (αj ) = (jT (αj )2 + jL (αj )2 )(1/2) can be
determined and are depicted in figures 79(g) for the disc and 79(f ) (only jc (αj )) for the squares.
The critical current densities in T - and L-directions are determined by jc,T = jT (αj = 0◦ )
and jc,L = jL (αj = 90◦ ), respectively. Remarkably, the observed angular dependence of jc
is not given by the vector superposition of the jc,T -component (αj = 0◦ ) and jc,L -component
(αj = 90◦ ). This would result in the elliptical angular dependence
jT (αj ) = jc,T sin(αj ),
jL (αj ) = jc,L cos(αj ),
2
2
jc (αj ) = (jc,L
cos2 (αj ) + jc,T
sin2 (αj ))1/2 ,
(75)
which is plotted as bold (jc ), dashed (jT ) and dashed–dotted lines (jL ) in figures 79(f ) and (g).
It was shown by a quantitative analysis of jc (αj ) that the deviations from an elliptical
angular dependence can be interpreted in terms of quasiparticle (QP) scattering-induced vortex
pinning (Jooss et al 1999, 2000a). The strong decrease of jL for increasing αj > 90◦ reflects
the additional scattering of the QP when an increasing current component starts to cross the
boundaries. The additional scattering leads to a depletion of the superconducting charge
density ns (ptr,T ) between the planes. This affects both current density components jT (αj )
and jL (αj ) and results in a stronger decrease than jL ∝ cos(αj ).
The presence of an order parameter suppression on small length scales was proven by
the observed Ambegaokar–Baratoff-like temperature dependence (Ambegaokar and Baratoff
1963) of jc,T (Jooss et al 1999). Additional evidence for a QP scattering pinning mechanism
and a scattering-induced suppression of the order parameter at APBs can also be derived from
the consideration of the observed magnitudes of jc,L and jc,T as a function of the defect width
rp : the QP scattering mechanism supports order parameter suppression on length scales which
are significantly smaller than the BCS coherence length ξ0 , which is the relevant length scale
for the δTc effect. This is consistent with the observations reported by Jooss et al (2000b); see
figure 80.
To summarize the results on APBs, they represent very effective pinning sites for correlated
pinning of vortices in HTS and can account for critical current densities jc,L up to 30% of the
depairing current (if both fp,T and lL are large enough). Remarkably, jc,T reaches values
which are typically observed in high-quality YBCO films on well oriented single-crystalline
substrates. For this large jc,T a small structural width of the APBs of rp = 0.7 nm is essential,
whereas a drop of jc,T to values 1011 A m−2 at 4.2 K is observed by introducing EDs with
rp ≈ 2 nm into the films. Via the visualization of the anisotropy of jc , an analysis of the size
of the flux filaments as a function of the defect width, the local magnetic field dependence
of jc,T (B) and a continuous mapping of jc (αj ), magneto-optical imaging has developed new
tools and standards for the analysis of flux pinning.
8.3. Surface pinning effects
Since the largest jc ≈ (1–5) × 1011 A m−2 at T = 4.2 K in HTS are usually observed in
highly textured epitaxial thin films, the question of a contribution of surface pinning effects
has been posed by several authors (McElfresh et al 1992, Schalk et al 1996, Jooss et al 1996a,
1996b). In general, a surface pinning effect in HTS could be questionable due to the lower
energy of vortex kink and double kink formation in anisotropic superconductors. However, a
quantitative estimate of the line tension of a vortex kink as a function of the wavelength shows
that in REBaCuO thin films (RE = Y, Dy, Sm, . . .) with moderate anisotropy γ = λc /λab ≈ 5
and thickness d λ, the kink energy which could induce a vortex jump into a neighbour surface
pinning site is of the order of the vortex self-energy and thus is unfavourable.
768
Ch Jooss et al
Figure 79. Angular dependences of jc at an array of parallel APBs: (a)–(d) flux patterns and
current streamlines of four squares with different angles αj between the sample edges and the
APBs. All squares are patterned from one YBCO film on SrTiO3 (106); d = 280 nm, width
W = 900 µm. All images are taken at µ0 Hex = 24 mT and T = 5 K. In (a) and (b) the MOL
has a defect which is visible as a dark line. (e) Flux and current distributions for a disc-shaped
YBCO film on SrTiO3 (106) with a radius of R = 1 mm at µ0 Hex = 28 mT and T = 5 K. The
quantitative analysis of the angular dependence of jc (αj ) for the four differently oriented squares
and of |jc |(αj ), jL (αj ) and jT (αj ) for the disc are shown in (f ) and (g), respectively. In addition,
the elliptical angular dependence following from equation (75) is plotted as bold and broken curves
in (f )–(g).
A second objection is that if the total critical current is caused by microscopic pinning
forces fpv in the volume as well as by surface pinning forces fps , there is now a simple way to
separate the two contributions. All complex issues of statistical summation may be taken into
account. An additive separation of the mesoscopic pinning force densities according to
jc = Fpv /0 + Fps (d)/0 = jcv + jcs (d)
(76)
can be justified, however, if the vortices do not feel the strongest pinning site only, e.g. if
the density of pinning centres in the volume is much larger than the length scale of the
surface pinning potential. Here, jcv denotes a thickness-independent contribution due to volume
pinning and jcs (d) a thickness-dependent contribution resulting from the surface pinning.
Current distributions in high-Tc superconductors
769
a)
|ψ |
APB
ξo
APB
b
x
6 nm
b)
|ψ |
APB
APB
ξ (ptr,T )
ξo
x
6 nm
Figure 80. A schematic diagram of the order parameter (OP) suppression due to (a) the δTc effect
and (b) quasiparticle scattering at a planar defect. For the δTc effect the relevant length scale for
a significant OP suppression is the BCS coherence length ξ0 . For the scattering mechanism, the
length scale ξ(ptr,T ) can be smaller.
In this section we summarize space-resolved studies of two surface pinning mechanisms:
firstly, vortex pinning due to a thickness modulation of the superconducting film; and secondly
vortex pinning due to an inhomogeneous proximity effect in YBCO/Ag bilayers.
8.3.1. Pinning by surface roughness. Due to the island growth mode, HTS epitaxial thin films
in general show a considerable surface roughness (Norton et al 1991, Raistrick and Hawley
1993, Ece et al 1995) with thickness variations d of the films between 10 nm and more than
100 nm (see figure 81). Naturally, the flux lines prefer positions at the smallest film thickness.
In typical epitaxial films the island size varies between 50 and 800 nm and thus the wavelength
l of the surface roughness may become smaller than the magnetic size SM ≈ 2λab of a vortex
oriented in the c-direction. Consequently, the change in the electromagnetic structure of a
vortex located at different surface positions has to be taken into account. The critical current
density due to surface pinning jcs has to be calculated by considering the change E of the
total vortex energy, thus giving
1 dE(d, d(r), l, λ) jcs =
(77)
,
d
dr
0
jcs
max
then becomes a function of the average film thickness d, the thickness variation
where
d, the wavelength l of the thickness variation and the magnetic size of a vortex SM (d) =
2λ(d) = 2(λ2ab + (2λ2ab /d)2 )1/2 which is larger than the bulk SM due to vortex widening at the
surface (Pearl 1964, 1966, Fetter and Hohenberg 1967). The total line energy of the vortex line
in equation (77) can be separated into three contributions E = ε0 d + εψ d + EBs , where the
first term represents the vortex energy in the unmodulated lower part of the film. The second
term is the condensation energy in the thickness-modulated upper part d < z < d + d of the
film, and EBs represents the electromagnetic energy of the vortex in the thickness-modulated
region. Since ξ l SM (d), the contribution of the condensation energy of the vortex
core εψ d can be approximated by considering the vortex as a point particle in the landscape
770
Ch Jooss et al
e)
Figure 81. AFM images of the surfaces of four YBCO films with increasing thicknesses of (a)
d = 140 nm, (b) d = 250 nm, (c) d = 290 nm and (d) d = 700. These films correspond to series
S1 in (e), where the roughness increases with the film thickness d. (e) The thickness dependence
of the mean square thickness variation S for the three series of films with different fabrication
methods. Note the different dependences of S(d) for the various series.
of a potential ∼d(r ), whereas for the electromagnetic contribution, the deformation and
delocalization of the electromagnetic vortex structure on roughness length scales l has to be
taken into account. If the d(r ) landscape is approximated by a cos(r ) potential and special
positions of the vortices at the minima and maxima of the potential are considered, the variation
of the electromagnetic vortex energy EBs is given by
EBs =
φ02
4πµ0 λ4ab
0
∞
r dr d cos
2πr
l
K12
2
R
R
r
2
+
K
,
0
SM (d) R 2
SM (d)
(78)
2 1/2
) . A quantitative evaluation of equations (77) and (78) is performed
where R = (r 2 + 2ξab
by Jooss et al (1996b) and is shown in figure 83(a).
The experimental separation of jcs from jcv requires a well defined modification of the
surface structure and film thickness d (determining jcs ) while leaving jcv constant. In the
following we summarize a magneto-optical study of YBCO films on SrTiO3 (100) substrates,
where the parameters l, d and d have been modified systematically (Jooss et al 1996a).
Therefore, 16 stripes with a width of W = 200 µm were patterned from only three series of
YBCO films S1, S2 and S3 (for pulsed laser deposition, see Habermeier (1991)). The thickness
of the films varied from 140 to 700 nm. Each strip was etched by ion milling in order to reduce
the thickness and to produce different surface roughnesses and therefore vary jcs and leave jcv
constant. The ion milling parameters were chosen carefully and oxygen annealing was done
afterwards to guarantee a constant jcv .
The surface structure of all strips was determined by atomic force microscopy (AFM)
(see
d-dependences of the mean square surface roughness S =
figure 81). Different
2
1/2
( N
could be realized for the series S1, S2 and S3 (see figure
i=1 (d(ri ) − d) /(N − 1))
81(e)). More details on the surface properties can be found in Jooss et al (1996a).
Current distributions in high-Tc superconductors
771
a)
c)
b)
Figure 82. A three-dimensional representation of the Bz -component of the flux distribution of a
250 nm thick YBCO strip (W = 200 µm) after ZFC to 4.2 K and application of µ0 Hex = 43 mT
(a) and 64 mT (b). The strip edges are located at the two maxima of Bz . MOL: EuSe. Spatial
resolution: 610 nm. (c) The thickness dependence of critical current densities measured for the
series S1, S2 and S3.
Figures 82(a)–(b) show a high-resolution measurement of the flux distribution of one
of the strips and figure 82(c) depicts the experimentally determined jc as a function of the
film thickness d. The magnetic flux distribution significantly deviates from a smooth Bz (y)
curve. An apparently statistical distribution of locations with high and small ∂Bz (y)/∂y
gradients are present (corresponding to regions with high and small jc ). Such structures in
the Bz (x, y) distribution may have different causes, such as kinetic roughening or instabilities
like flux avalanches (see section 5.3). A statistical analysis of the dependence of the maxima
in ∂Bz (y)/∂y on Bz shows a clear trend: the highest (∂Bz (y)/∂y)(Bz ) ∝ jc (B) values occur
at the matching field, which corresponds to the measured wavelength of the surface roughness
(Jooss et al 1996a). At the matching fields, a maximum pinning force Fp (B) is obtained
because the spacing of the vortex lines fits the roughness wavelengths of the surface (Ami and
Maki 1975, Martinoli 1978).
Additional evidence for the presence of a contribution of surface pinning to the observed
jc (d) can be found in figures 82(c) and 83. Figure 83 presents the theoretical thickness
dependence of jcs as a function of d and thickness modulations between d = 10 and 80 nm.
For d > 2λab the critical current density due to surface pinning follows a jcs ∼ d/d law,
while for small film thicknesses d < 2λab the critical current is reduced due to vortex line
widening. With these theoretical results the experimental results presented in figure 82 for the
critical currents of the three series of films as a function of d can be interpreted consistently
by choosing appropriate values for the volume critical currents jcv (which is the only free
parameter in the fits and is equal within each series, but differs between different series; all
other parameters d, d and l are determined by AFM). The results of a comparison between
calculated and measured critical currents are shown in figure 83(b) for the three series of films
investigated. It is of interest to note that the largest critical currents for all series are observed
for d ∼ 2λab as predicted by the theoretical model. The series with the largest thickness
modulation reveals the largest critical current densities due to surface pinning.
772
a)
Ch Jooss et al
b)
Figure 83. (a) The theoretical thickness dependence of the critical current density for d = 10–
80 nm and l = 500 nm. (b) Calculated and measured jc for all three series 1, 2, 3. The increase
of jc of 30% for series 2 with decreasing d and increasing roughness is clearly demonstrated. The
volume critical currents jcv were 1.8 × 1011 (series 1), 2.6 × 1011 (series 2) and 0.1 × 1011 A m−2
(series 3). jcv represents the only fit parameter; all other parameters describing the surface topology
are determined by experiment.
Prozorov et al (1998) stated that for thin films with thickness d < λ the supercurrent due
to surface pinning would also be mainly concentrated within a small surface layer. This would
lead to a 1/d dependence of the measured sheet current Jc in contrast to the observations
reported above. In their calculation of the vortex curvature along the z-axis, the vortex line
widening in thin films was not taken into account.
In conclusion, the contribution of surface roughness pinning to the total critical current is
small (typically ≈10%) and the large jc of epitaxial HTS films are not mainly due to surface
pinning effects. Numerical values for fp derived from the variation of the vortex energy
according to equation (77) are fp = 6×10−6 –5×10−5 N m−1 for typically observed thickness
modulations of 10–80 nm in epitaxial YBCO films. These pinning forces are comparable to
pinning forces due to elastic interactions with screw dislocations (fp = 5 × 10−6 N m−1 ) and
exceed the elastic interaction with edge dislocations, where fp = 5 × 10−7 N m−1 (Ullmaier
1975). However, the pinning force per unit length due to correlated disorder can reach much
higher values in HTS; in particular one has fp up to ≈ 10−3 N m−1 transverse to antiphase
boundaries and fp up to some 10−4 N m−1 at columnar defects in YBCO. Only for one film
of series S2, where the roughness was artificially increased up to d/d = 0.36 is a large
contribution of surface pinning observed (jcs up to 30% of the total jc ). The highest jcs are
expected for roughness wavelength l ∼ 2λab and average film thicknesses of d = 2λab
together with a maximum thickness variation d.
8.3.2. Pinning by the inhomogeneous proximity effect. In technical applications, HTS
interfaces between superconducting and normal-conducting materials (SN boundaries) play
an important role. In particular, the superconducting wires in tapes are embedded in a silver
matrix and it is of interest to investigate the effect of the silver matrix on the critical currents of
HTS wires. In addition, layers of a noble metal are often used on HTS films as contact pads for
the current lead and also as protective coatings against degradation of the films. In magnetooptical experiments on YBCO films with high-quality Ag or Au coatings a local enhancement
of jc in the coated area is regularly observed (Forkl et al 1996a). A systematic study of the jc enhancement at YBCO/Ag bilayers as a function of the superconductor and normal-conductor
film thicknesses ds and dn , respectively, was performed by Kienzle et al (2001).
Current distributions in high-Tc superconductors
773
Figure 84. The flux penetration into a YBCO strip (width w = 500 µm) covered by a silver film
on the left side of the strip (ds = 125 nm, dn = 295 nm). The field is applied normal to the film
after ZFC to 42 K.
Figure 84 shows the flux penetration into YBCO thin film strips, where the left side is
covered by the thermally evaporated Ag. In the silver-coated part of the strip the flux penetration
is much lower than in the uncoated YBCO strip, indicating that the critical current density is
enhanced on the silver-coated side. By applying this space-resolved difference method, the
jc -difference between the coated and non-coated areas of one YBCO film is determined with
a very high precision within one magneto-optical measurement. Artefacts (due to sample
processing or measurement errors) which could occur when comparing different samples in
different measurement runs can be ruled out. A sketch of the current pattern according to the
Bean model is compared with the measured current distribution in figure 85. In the silvercoated region the critical current is ≈50% larger than in the uncoated part of the YBCO film.
For a detailed discussion of the current pattern and the d-line structure, see section 6.4.
One possibility for the origin of the jc -enhancement is the shunting of the supercurrents
by the metallic layer. However, this effect alone cannot explain the observed jc -enhancement
because the additional current which flows directly on top of the superconductor increases
the Lorentz force on the flux and can be maintained in the bilayer only if the counteracting
pinning force is also increased. Consequently, a larger pinning strength is required to explain
the observed jc -enhancements.
Measurements of the increase jc as a function of the thickness of the superconductor ds ,
shown in figure 86(a), clearly show the nature of this effect as a surface effect. The dependence
of jc on dn gives strong evidence of its relation to the proximity effect, where the condensate
774
Ch Jooss et al
a)
b)
c)
Figure 85. (a) The magnetic flux distribution at the boundary region of an uncoated (left side)
and an Ag-coated YBCO film (right side) after ZFC to 4.2 K and application of µ0 Hex = 60 mT
(ds = 97.5 nm, dn = 345 nm). The full lines correspond to current flow lines obtained by inversion
of the Biot–Savart law. (b) A greyscale image of the absolute value |j (x, y)| of the current density
corresponding to the flux distribution in (a). (c) A schematic diagram of the current density
distribution within a strip with regions of different critical currents j1 and j2 in the framework of
the Bean model. The d-lines are depicted with larger line thickness.
of the superconductor penetrates the normal metal. The largest increase of jc is obtained for
a silver film of 295 nm at an YBCO thickness of ds = 120 nm. Due to the proximity effect,
Cooper pairs of the HTS can penetrate the metallic layer with penetration depths in the range
of ∼100 nm (Deutscher and De Gennes 1969). In the vortex state of a superconductor/metal
interface, the vortices extend virtually into the metallic layer due to this effect. Consequently,
a spatial variation of the transmission amplitude W (r) of the Cooper pairs (due to a varying
interface quality, precipitates and inhomogeneities including surface roughnesses) creates a
pinning potential for the vortices. The critical current density due to this surface effect is
approximately given by
1 dE(r)
1 dE dW (r)
jcs =
=
.
(79)
d0 dr
d0 dW dr
The calculation of the line energy En (r) of a vortex extending virtually into the normalconducting metal is based on the ansatz for the order parameter
ψ(r, z) = f (r)g(z)eiϕ ,
(80)
where f (r) describes the radial distribution of ψ(r, z) and is approximated by equation (4)
according to Clem (1975). The function
W
1
g(z) = ψ0
(z − dn )
(81)
cosh
cosh(dn /ξn )
ξn
gives the dependence of ψ parallel to the vortex line which is assumed to lie perpendicular
to the SN boundary (Golubov and Kupriyanov 1994). ψ0 denotes the order parameter in the
superconductor and W describes the jump of the order parameter at the SN boundary. W 2
corresponds to the transmission coefficients of the Cooper pairs at the SN boundary. ξn is
the so-called proximity length in the normal conductor. For the evaluation of equation (79),
see Kienzle (2001).
Current distributions in high-Tc superconductors
775
a)
b)
d)
c)
Figure 86. (a) The enhancement jc of the critical current density in the silver-coated YBCO film
as a function of 1/ds ; ds denotes the thickness of the superconductor. (b) The change jc in the
dependence of the thickness dn of the silver film for different ds . (c) The dependence of jc on dn
for different proximity lengths ξn (W̃m = 0.04, x = 20 nm). (d) Comparison of the experimental
quantity jc ds with the theoretical prediction as a function of dn for the specimen of figure 84.
The fit parameters were W̃m = 7.46 × 10−3 , x = 27.1 nm. Since jc = 0 for dn → 0, an
additional surface pinning effect due to the modification of the YBCO surface leading to an offset
current density of jcs = 0.25 × 1011 A m−2 has to be assumed.
Numerical results for jc are presented in figure 86 as a function of dn with ξn as a
parameter. The superconducting and microstructural parameters used for this calculation are
κ = κab ∼ 100, λL = λab ∼ 150 nm, ξn as given in the plots, W̃m ∼ 1/25 (Kupriyanov
and Likharev 1990) and l ∼ 20 nm. The numerically determined critical currents jc due to
pinning by the inhomogeneous proximity effect are of the correct order of magnitude of several
1010 A m−2 . jc increases linearly for dn < ξn and approaches a saturation value for dn > ξn .
The saturation values increase with increasing ξn linearly and with increasing W̃m quadratically.
A comparison with the experimental results is shown in figure 86(d), where the experimental
quantity jc ds versus dn is presented. Furthermore, an additional term, jcs = constant, has
been added to the proximity term which describes the finite jc , extrapolating ds → 0. A
possible explanation is an additional microstructural pinning at the inhomogeneously degraded
surface due to precipitates and plasma oxidation. The proximity length is estimated to have a
value of ξn ≈ 350 nm, which is surprisingly large compared to the small coherence length in
HTS (and should be confirmed by further experiments).
In summary, as shown by magneto-optical experiments, the presence of a noble metals next
to HTS cannot lead just to shunting effects on length scales of several 100 µm (see section 7.5).
In geometries where the additional current in the metal is acting via the Lorentz force on the
vortices in the superconductor, an enhancement of the vortex pinning forces has to occur. The
results presented in this section clearly support an additional pinning by an inhomogeneous
proximity effect. This finding together with the relatively large proximity length determined
is interesting in particular as regards the development of superconducting cables based on Ag
776
Ch Jooss et al
tapes or CC, since a part of the supercurrent flows in the normal metal and, in addition, the
surface pinning effect improves jc .
9. Summary and conclusions
Having considered the magnetic flux and current patterns of various HTS with different
geometry, microstructure, texture and pinning sites together with the related physical
mechanisms, it is now necessary to summarize and to discuss what we already understand
and, in particular, what we do not understand yet. This is not an easy task and we hope that
the reader will excuse us for presenting only a selection of topics in the following. One of the
key questions that we want to consider here is what is really new in the understanding of the
spatial current distributions of HTS.
One of the most important factors making it necessary to apply space-resolved
measurements of the supercurrents to HTS is that they exhibit many more current-disturbing
spatial inhomogeneities compared to metallic superconductors. The higher complexity of
the defect structure and chemistry of the oxide superconductors gives rise to a much richer
variety of microstructures. Since the superconducting state in HTS occurs near an insulator–
metal transition, the influence of structural distortions on the superconducting state and the
local current distribution can be drastic. One of the most prominent examples is provided
by GBs: there is much experimental evidence that the disorder introduced by dislocations
in LAGBs or non-stoichiometric structural units in HAGBs is responsible for a strong hole
depletion and causes an exponential decrease of the grain boundary critical current with
increasing misorientation angle of the adjacent grains. Consequently, grain boundaries, if
present, strongly dominate the magnetic flux and the current pattern in HTS samples and
represent one of the most serious current-limiting defects in HTS. While first steps have been
made in the investigation of the spatial current distribution through GBs and GB networks, a
full understanding of the current density distribution as a function of the GB angle, the value
and the angle of the magnetic field, the electric field, the size and geometry of the sample and
the arrangement of GBs is far from being achieved.
Due to their current limiting, GBs have a strong impact on the development of large-scale
high-current applications of HTS. Magneto-optical experiments have strongly contributed to
the analysis of current-limiting features like cracks, GBs and GB networks and precipitates
in technical materials, such as CC, tapes and melt-textured bulk samples, developed for
applications.
Another microscopic property influences the current-carrying state of cuprate HTS: the
d-wave nature of the superconducting state with nodes in the superconducting gap function
strongly assists the scattering of quasiparticle excitations at various kinds of defect and therefore
leads to a spatial variation of the superconducting charge density ns on nanoscales in disordered
material. This creates a new type of pinning potential for vortices which is not present in
superconductors with isotropic gaps. Additionally, a spatial variation of ns directly influences
the value of the depairing current and represents a further contribution to the observed spatial
variations in jc as shown in magneto-optical studies with arrays of antiphase boundaries.
Like the strong thermal fluctuations, the anisotropic crystal structure of HTS represents
a material property which has a strong impact on the transport properties. First of all,
the anisotropy of the intrinsic superconducting parameters is directly related to anisotropic
current distributions. More fundamental are the consequences for the vortex structure, since
the anisotropy strongly reduces the line tension of the vortex lines and changes the vortex
structure as a function of its orientation within the crystal. This brings up new depinning
modes of vortices by the development of vortex kinks and double kinks. Simultaneously,
Current distributions in high-Tc superconductors
777
the anisotropy together with large thermal fluctuations of the condensate and quasiparticle
excitations enhance strongly pronounced metastable features of the critical ‘state’, like flux
creep and vortex depinning processes, already present far below the irreversibility line. Global
transport experiments reveal that the interplay between thermal, elastic and pinning energies
forms different solid and liquid vortex phases. However, up to now, only a few space-resolved
measurements for vortex phase transitions and the coexistence of different vortex phases have
been performed.
In view of this, it is somewhat surprising that the current distributions as observed by
means of magneto-optics are in many cases very well understood on the basis of Bean-like
quasi-equilibrium critical state models. The Bean model describes the current pattern as a
vector field with constant magnitude, resulting from an equilibrium balance of the pinning
and driving forces on flux lines. Taking into account the correct boundary conditions at the
sample’s surfaces and inner interfaces, the formation of current domains with uniform parallel
current flow, separated by discontinuity lines, is very well described in a qualitative manner.
The extension to samples with areas of different jc -values even gives a qualitatively correct
idea of the complex patterns of d + - and d − -lines being formed.
However, for a full quantitative understanding of the current distribution, the time
development and consequently the electric field distribution during the formation of a particular
magneto-optically observed flux pattern has to be taken into account. In particular, in regions
with inhomogeneous jc such as GBs, it is shown experimentally that the electric field pattern
and the related time decay of the current pattern are strongly inhomogeneous. In such cases, jc
at different positions is measured at different electric field criteria. Special sample geometries
and current obstacles assist the formation of exponentially large electric field peaks. Theory and
experiment suggest that the structure of the Bean discontinuity lines evolves into current domain
boundaries with finite width which depend on the time window of the experiment. Experiments
on these issues are still preliminary. This is also the case for the investigation of the flux and
current patterns in different vortex phases, instabilities and turbulence. The development of the
magneto-optical technique to a full quantitative imaging technique for current densities with
high spatial and high time resolution opens up new possibilities for nice future experiments.
Acknowledgments
This work would have been impossible without the contributions of various colleagues to the
experiments described here. We are sorry that it will not be possible to mention everybody
here. The authors are grateful to H-U Habermeier and B Leibold for the preparation of various
YBCO films, to T Haage and J Zegenhagen and H-U Habermeier for wonderful cooperation
in investigating vicinal films, to R Spolenak for the irradiation of the substrate surfaces with
the FIB, to G Cristiani for YBCO films and bicrystals, to A Leenders, H Walter, B Bringmann,
M P Delamare and H C Freyhardt for cooperation concerning melt-textured YBCO. Gratefully
acknowledged also are the generous offer of U Krebs to prepare the laser ablation in his
laboratory, the cooperation with B Holzapfel and B de Boer on the RABiTS tapes and the
support in investigations on IBAD-coated conductors of J Dzick, S Sievers and H C Freyhardt.
Much thanks to Th Schuster, E H Brandt, R Warthmann, M Kienzle, A Forkl, M Becher,
T Dragon, M Kotz and B Ludescher from the Max-Planck Institut für Metallforschung for
several figures and magneto-optical images. We are also grateful to Q Wang for figure 5(b),
to T Johansen and his co-workers for figures 8, 14(a)–(d), 36, 51, 52, 70, to M Indenbom for
figures 15, 31–32 and 74, to J Mannhart for figure 46(b), to D C Larbalestier, A A Polyanskii,
M Feldmann and their co-workers for figures, 61, 62, 63, 66–68, to G Crabtree, U Welp,
V K Vlasko-Vlasov and their co-workers for figures 34, 35 and 69, to A Leenders for
778
Ch Jooss et al
figure 54, to L O Kautschor for figures 60(b) and (c), to T Haage for figure 77(a), to M
Koblischka for figure 50 and to H-U Habermeier and B Leibold for figure 83. This work was
partly supported by the TMR programme SUPERCURRENT of the EU under contract no
ERBFMRXC98-0189.
Gratefully acknowledged also are the strong input from and helpful discussions with
T Johansen which contributed substantially to the improvement of the manuscript. Helpful
criticism and suggestions came from J Hoffmann, K Guth, V Born, H Jarzina and C Herweg.
We also gratefully acknowledge the general support of H C Freyhardt for this work.
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