Supercond. Sci. Technol. 11 (1998) 333–357. Printed in the UK
PII: S0953-2048(98)74607-X
TOPICAL REVIEW
Current transfer and initial dissipation
in high-Tc superconductors
M Prester
Institute of Physics, POB 304, HR-10 000 Zagreb, Croatia
Received 28 November 1997
Abstract. Various aspects of the problem of current transfer in high-Tc
superconductors (HTSs) are reviewed. The spatial inhomogeneities of various
types are identified as a primary cause of non-uniformity of both normal currents
and supercurrents in real samples of HTSs. The role these inhomogeneities play in
transport features of the samples is discussed. The case of grain boundaries in
polycrystalline samples is elaborated in detail. The local structural and transport
properties of isolated grain boundaries are first reviewed and then integrated into
the knowledge of global (macroscopic) charge transport. The paper emphasizes
the common ingredients characterizing the transport in various forms and families
of HTS samples in small magnetic fields. The phenomenon of percolation is
identified as the most obvious one and is shown to dominate a large number of
observations covered by this report. The experimental results focused on by this
report elaborate primarily the problems of critical currents, initial dissipation and
current–voltage characteristics, penetration depth, resistive and metal–insulator
transition, resistance noise and magneto-optical studies of current paths. Various
models for current transfer (disordered bonds, brick wall and railway switch) are
also reviewed and discussed.
4.2.
Contents
1.
2.
3.
4.
Introduction
Current transfer in normally conducting and
superconducting systems
Current transfer in high-Tc superconductors:
local aspects
3.1.
Structural features of grain boundaries
and Josephson coupling
3.2.
Mechanisms of Josephson coupling
across the grain boundary
3.2.1.
Ic (T )
3.2.2.
Rn and Ic Rn product
3.3.
Tunnelling across the grain boundary
Current transfer in high-Tc superconductors:
global aspects
4.1.
Percolation in electrically heterogeneous networks: disordered-bonds
model
4.1.1.
Critical currents
4.1.2.
Initial dissipation and current–
voltage characteristics.
4.1.3.
Penetration depth
4.1.4.
Resistive transition
4.1.5.
Metal-insulator transition
4.1.6.
Resistance noise
c 1998 IOP Publishing Ltd
0953-2048/98/040333+25$19.50 333
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335
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340
340
341
342
343
346
347
347
347
5.
Microstructure-oriented models for
macroscopic current transfer: brick-wall
and railway-switch
4.3.
Visual inspection of percolative
current paths
4.3.1.
Magneto-optical studies
of current paths in highTc superconductors
Discussion and conclusion
References
348
349
350
353
354
1. Introduction
What do the current paths in high-Tc superconductors
(HTSs) really look like on various spatial scales and
how are these paths determined by local properties?
The answers to these at first glance rather technical
questions seem to have important consequences not only
for applications of HTS materials but also for many
aspects of basic understanding of these systems and of the
phenomenon of superconductivity in general. Knowledge
of the average distribution of supercurrents in classical
superconductors, usually depicted in many textbooks [1]
by phrases such as ‘thin layer of surface current’ and
‘bulk current of critical state’, although equally applicable,
in principle, to HTSs, provides, however, insufficient
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M Prester
insight into the details of local current distributions.
The local transport properties play a dominant role in
these distributions, being also responsible for global
(macroscopic) charge transfer. A unified picture of both
dissipative and non-dissipative current paths has not been
entirely formulated yet. The purpose of this paper is
therefore mostly to bring together those ideas and results
which deal, either explicitly or implicitly, with the problem
of charge transfer in the specific transport medium of
HTSs. There are certainly many different reasons why
this medium may be considered, from the standpoint of
transport phenomena, as a rather specific one. Moreover,
there are various reasons why, in turn, the various forms
that the real samples of HTSs are prepared in (e.g. single
crystals, sintered polycrystals, silver-clad tapes, thick and
thin films), as well as various families of HTSs, reveal their
own specialties, particularly in their transport properties.
In this paper we will focus, however, only those aspects
of charge transfer which reflect a ‘generic’ problem of this
class of new superconductors, i.e. the absence of (or the
difficulties with) a true long-range order. This problem
concerns both the structural order, defective as a result
of structural inhomogeneities on various scales, as well
as the superconducting order, limited by the intrinsically
small coherence length characterizing these systems. A
simple but important consequence of a combined effect
of both types of defective order, which characterizes,
although in a specific way, all forms of HTSs, is that
the vector of (super)current has to be considered as a
function defined on a local level, differing significantly,
in its magnitude and direction, from its spatial averages.
Therefore, instead of uniform currents of a hypothetical
perfectly long-range-ordered system the realistic current
paths in HTSs are non-uniform in principle and are subject
to complicated meandering and multiple local branching
(obeying, however, charge conservation) in order to achieve
energetically the most favourable current distribution, i.e.
a distribution which minimizes overall dissipation in the
sample. More specifically, in HTSs one could identify
at least three groups of mutually related phenomena, all
characterized by the absence of long-range order, which
underlie inhomogeneous currents on various spatial scales:
defective structural order (spatially periodic or aperiodic
compositional variations, local deviations from average
structure, twin boundaries etc in single crystals and epitaxial
films, grain boundary features in various polycrystalline
bulk and thin film forms of HTS), thermodynamically
competing and possibly coexisting locally ordered phases
(characterized by superconducting, normally conducting
or magnetic ground state) and, in applied magnetic
fields, disordered vortex lattices (characterized by an
extraordinarily rich phase diagram).
Of course, the
presence of extrinsic defects such as cracks, segregated
secondary phases, voids and impurities leads to similar
effects on current transfer and may be considered as an
extrinsic category of inhomogeneity.
The present review focuses however, on the problem
of inhomogeneous currents in HTSs from a more restricted
viewpoint. It mainly deals with the problem of dissipative
and non-dissipative currents in polycrystalline (textured
334
Figure 1. SEM image of the microstructure of a
polycrystalline YBCO sample. As well as grain boundaries
and some intragranular defects, extrinsic defects (voids,
cracks, secondary phases, etc) can be easily identified.
Courtesy of Dr J Mirkovic.
or isotropic) HTS samples in the absence of or in
small applied magnetic fields, focusing primarily those
features of current transport in HTSs which represent
direct consequences of, or are directly related to, their
spatially heterogeneous nature. Figure 1 illustrates the
microstructure of polycrystalline HTSs. There are two
reasons for the emphasis on various polycrystalline forms:
firstly, these forms of samples are the most widely
investigated so far, and, secondly, the charge transport in
a relatively simple weak-link network of polycrystalline
sinterates and films as well as of silver-clad composites,
although interesting and important by itself, may also be
considered as a qualitative starting model for studies of
transport in more complicated systems (intrinsic transport
in single crystals) or in different and more advanced but
still analogous physical situations (dissipative excitations
in fluxoid lattice). The restriction to polycrystalline forms
defines simultaneously the mesoscopic (i.e. micrometre)
spatial scale as the scale that the present review primarily
applies to. It should also be noted that the results of
measurements on various HTS families (i.e. YBa2 Cu3 O7−x ,
Bi2 Sr2 Ca2 Cu3 O10+y and Bi2 Sr2 Ca1 Cu2 O10+y , hereafter
YBCO, BSCCO-(2223) and BSCCO-(2212), respectively)
are reviewed mainly from the standpoint of those features of
current transfer which are considered common for samples
belonging to all families. The question of the remarkable
differences between the families or samples in many cases
has not been systematically analysed in this review. For this
aspect of the current transfer problem one should consult
the cited original papers.
2. Current transfer in normally conducting and
superconducting systems
Transport of electrical charges (i.e. electrical current) obeys
different physical laws in the normal and superconducting
phases of a conducting material that the transport is
studied in. In the normal phase the vector of local
Current transfer and initial dissipation in HTSs
dissipative current is related to local electric field, a
‘driving force’ for charge redistribution, by a rather
complex Boltzmann (transport) equation [2]. This generally
non-linear tensor equation has to take into account, as
well as the properties of the conduction band at the
Fermi surface and the properties of the surface itself, all
possible scattering processes taking place in the sample.
In the superconducting phase the non-dissipative current
is a feature of stationary solutions of the Ginzburg–
Landau equations [1] which one may associate with the
sample and its boundary conditions. These equations
are in one-to-one correspondence with a microscopic
description of the superconducting state. The spatial
changes of the order parameter and the local value of
the magnetic vector potential play formally the role of a
driving force for stationary non-dissipative currents [1].
Transport in both phases, normal and superconducting,
strongly depends on the concentration of various types
of defects or, generally, on disorder. The influence
of disorder is usually accounted for in reformulated
(renormalized) forms of characteristic quantities, such
as mean free path (normal state) and coherence length
(superconducting state). However, in cases when defects
form a spatially organized (sub)structure(s) this approach
becomes questionable and, depending mainly on the
representative sizes of these substructures and on the scale
of mean free path and/or coherence length, the charge
transport may be better described by modelling the sample
as an electrically heterogeneous medium. Knowledge of the
current distribution depends now not only on Boltzmann
or Ginzburg–Landau equations describing one or both
subsystems but also on processes at their interfaces. The
latter processes may contain new physical ingredients
that are crucial for global electrical conduction. Grain
boundaries in HTSs represent precisely the defects which
belong to the latter category. In order to discuss the
macroscopic current transfer in HTSs we therefore first
review the properties of grain boundaries because of their
pronounced role in local aspects of the problem of charge
transport.
3. Current transfer in high-Tc superconductors:
local aspects
We first summarize the knowledge of structural features of
grain boundaries in HTSs, particularly those which were
found to be relevant for the transport of supercurrents, and
then specify their role in the problem of distribution of
supercurrents. The problem of normal transport across the
boundaries will be elaborated as well.
3.1. Structural features of grain boundaries and
Josephson coupling
The presence of grain boundaries in various forms of
polycrystalline samples is, in many cases, just a natural
consequence of their preparation from powdered precursors:
the conventional grain growth which takes place at elevated
temperatures increases the average grain size and alters
its distribution but does not eliminate inter- and intragrain
boundaries. There are certainly many other reasons and
mechanisms responsible for the very existence of the grain
boundaries in HTSs [3]. As is the case with other
complex structures, defects of many kinds limit the spatial
range of perfect structural order in HTSs. As well as
the usual structural defects (point defects, dislocations,
stacking faults, cracks etc) the defects in HTSs may have
also several specific sources: lack of a congruent melting
point in phase diagrams of constituent components [4],
disordered diffusion of oxygen inducing structural reordering or even real structural transitions in some HTS
systems [5], metastability of the locally disordered oxygen
sublattice [5, 6], incommensurate modulation along certain
crystallographic directions [5, 6] etc. Given the defects,
the grain boundaries (and single-crystal twin boundaries)
represent the energetically favourable response to the
increase of elastic energy introduced by the presence of
defects: the stress energy accumulated in the boundary
is usually smaller than the energy of homogeneously
distributed stresses inside the grain [7]. A grain boundary
is therefore a planar defect, separating the two adjacent
grains which have been rotated, i.e. tilted or twisted, with
respect with one another. Knowledge of the structural data
of grain boundaries in HTSs has been systematized in a
recent review by Babcock and Vargas [3].
The boundaries are designated by the type and amount
of misorientation of the two abutting grains: the grains
mutually rotated (tilted or twisted) by θ around the direction
[hkl] define the boundary ‘θ[hkl] (twist or tilt)’. If one
extrapolates the knowledge of grain boundary features
valid in other materials [7] to HTS grain boundaries [8],
it may be expected that various misorientation angles
are not equally probable, i.e. that some misorientation
angles are consistent with local minima of the crystal
energy of the system of grains. If free rotation of
grains–crystals is allowed, the low-energy structures occur
in cubic material whenever a coincident site lattice is
produced [9]. The latter lattice (or, better, bilattice)
comprises the common sites of interpenetrated original
lattices extended on both sides of a boundary. The
boundaries consistent with high coincidence (i.e. low value
of the ratio of coincidence lattice unit cell and unit cell
of original lattice, known as the coincidence index ζ )
have indeed been found to be favourable in a variety
of cubic materials [9, 10]. The investigations performed
on various HTS families [11–14] (YBCO, BSCCO) and
forms (those which allow free rotation in the process of
formation such as flux-grown crystals and polycrystalline
films) reveal peaks in misorientation angle histograms
at angles consistent with low ζ , confirming the general
applicability of the coincidence sites scheme in HTSs as
well. Macroscopic current transfer in HTSs is substantially
limited, in magnitude and spatial distribution, by the
current capacity of the grain boundaries. The orders of
magnitude higher critical current density of single crystals,
compared with the current density in polycrystalline forms
of HTS compounds, has been naturally interpreted as a
consequence of degraded superconducting properties of the
boundaries [15]. The nature of the degradation was, and
in some sense still is, a topic of much controversy [3].
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M Prester
The early expectations that the grain boundary segregation
of secondary phases could be a cause of current capacity
degradation indeed met some experimental support [16–19]
but there is now generally a consensus, primarily because
a large number of boundaries investigated so far can
be simultaneously clean and degraded, that the boundary
precipitates do not play a central role in grain boundary
degradation. The presence of secondary phases at the
boundary would certainly be detrimental to current transport
but the concentration of such boundaries in properly
prepared samples does not dominate, at least, over the
fraction of clean boundaries. However, in cases when
the presence of a substantial amount of other phases is an
intentional processing parameter, such as a liquid phase
during the melt texturing of YBCO bulk samples [20, 21],
the traces of these phases at the boundaries, together
with cracks, may indeed represent the main obstacles for
local non-dissipative transport. In more general cases,
which equally apply to all HTS systems, the current
capacity of a clean boundary is primarily determined by
its misorientation angle and its type. This dependence is,
however, quite a complex one and a detailed knowledge of
transport properties of various boundaries, correlated with
their structural and compositional features, is necessary
in any attempt to answer questions of the nature of
the grain boundary degradation. The direct experimental
studies of transport in the selected boundaries certainly
provide the most complete insight into this problem. The
investigated boundaries were either epitaxially grown on
tailored bicrystalline substrates or selected as a naturally
grown boundary in the appropriately prepared samples,
with a controlled misorientation of the adjacent grains
being achieved. The spectrum of possible observations
is representatively covered by the work of the IBM
group on individual YBCO boundaries grown on SrTiO3
bicrystals [22–26]. The studied geometries were θ[001]
tilt, θ[100] tilt and θ[100] twist. In the first of the chosen
representative boundary geometries the c-axes (tilting axis)
of neighbouring grains are parallel while in the latter two
the c-axes are mutually tilted or twisted around [100]. The
I –V characteristics were measured for currents traversing
the boundaries, allowing knowledge of the critical current
density and dynamical resistance to be acquired. The results
were scaled by corresponding measurements of the same
quantities but involving only the intragranular currents.
The most general conclusion from these measurements was
that for all but very small misorientation angles (θ <
5◦ ) the densities of critical currents across the boundary,
Jc (gb), are substantially smaller than those characterizing
the intragranular I –V characteristics, Jc (g) [22]. In the
range 5◦ < θ < 20◦ a rapid decrease (i.e. Jc (gb)/Jc (g) ∝
1/θ ) was detected while for higher misorientation the
saturation in Jc (gb) takes over, usually at a level 2 orders of
magnitude below the corresponding Jc (g) value [22]. It was
immediately clear that the intrinsic anisotropy of the layered
HTS system was not a principal cause of critical current
degradation: almost the same qualitative behaviour were
detected in all three representative geometries in spite of the
very different constraints each of them obviously imposes
on anisotropic transport of supercurrents. Hence, the
336
degradation has to be related to a suppression of the order
parameter (superconducting gap) in the boundary and/or its
neighbourhood. This attribution invokes, however, several
important questions. The first one is whether weakened
flux pinning or Josephson coupling (the suppression could
be accompanied by both) underlie the critical current
degradation. The latter dilemma has been resolved in
favour of Josephson coupling. There could be a number of
arguments supporting this conclusion [22] but the following
are certainly the most convincing ones: flux quantization
by grain boundary loops [27], observation of Fraunhofer
diffraction patterns on bicrystalline boundaries [28, 29],
observation of Shapiro steps and substeps originating from
boundaries exposed to microwave irradiation [30] and
the existence of operable SQUID devices based on grain
boundary junctions [23, 28, 31]. Some of these features are
shown in figure 2. The transport properties of the grain
boundaries can therefore be best described by the physics
of Josephson effect [1, 32] which relates, for example,
the experimental critical current Ic of a boundary to the
weak intergrain coupling energy Eg , Ic = (2e/h̄)Eg .
Assuming weak coupling across the boundaries there are
two additional questions that naturally arise: what is the
precise character of the weak coupling and what is its
microscopic and structural origin?
Related to the first question, concerning detailed
knowledge of the Josephson coupling character, it is
relevant to note that the boundaries are known [33] to
fit nicely the behaviour of resistively shunted Josephson
junctions (RSJs) [33, 34], and, as far as intergrain
supercurrent transport is concerned, are colloquially
termed ‘weak links’. This term, however, does not
a priori favour any of the traditional weak coupling
mechanisms [35], such as tunnelling of Cooper pairs [36]
(i.e. superconductor–insulator–superconductor (SIS) model)
proximity effect [37] (i.e. superconductor–normal metal–
superconductor (SNS) model) or ‘point contacts’ (narrow
constrictions) [37] all of which could, in principle, account
for the documented Josephson behaviour of the boundaries.
The relevance of these mechanisms will be discussed later.
Now we discuss the second remaining question, that of
the microscopic origin of order parameter suppression in the
boundary region. The formulation of an appropriate model
was the subject of much investigation. The primary goal of
the required model should be, for example, to interpret the
dependence of Jc (gb) on misorientation angle and apparent
division of the boundaries into two current-capacity classes,
low-angle (strong linked) and high-angle (weak linked)
ones. The existing models all stem from the detailed
knowledge of the structural properties of the boundaries
and general understanding of the grain boundary [22] as
a plane array of dislocations [38]. Indeed, transmission
electron microscopy studies of boundaries of various HTS
systems showed that boundaries accommodate regularly
spaced dislocations [23, 39]. In the energy representation
the dislocations are equivalent to an inhomogeneous strain
field characterized by locally and periodically increased
strain energy. The size of the strained region is defined
by radius of the dislocation core, rm . The core radius
is a specific quantity of the involved crystal lattice.
Current transfer and initial dissipation in HTSs
the simple relationship [39]
a)
Voltage (mV)
-0.02
Jc (gb)
2rm
=1−
θ
Jc (g)
b
-0.04
-0.06
-0.08
-0.10
-60
-40
-20
0
20
40
60
Magnetic field (Oe)
b)
Voltage (µV)
16
12
8
4
0
-5
-4
Magnetic field (Oe)
Figure 2. The grain boundary as a Josephson junction
(from [28]). The boundary was produced by laser
depositing YBCO on an yttria-stabilized zirconia bicrystal
substrate. The misorientation angle of the boundary was
32◦ . (a ) Josephson interference pattern of the single weak
link and (b ) the d.c. SQUID performance of a device
designed from thisboundary [28].
Taking into account the closeness of the antiferromagnetic
(electrically isolating) phase of HTS in the corresponding
phase diagrams [40], the local strain in the core could
be responsible, provided that the core’s spatial scale
overcomes coherence length of the compound, for local
suppression of the order parameter. In addition to rm , the
other two important lengths are the Burgers vector [38], b,
which defines the dislocation network, and the separation
between the dislocations, d. The latter two quantities
are not independent and, in the case of a symmetric tilt
boundary and the limit of small tilt angles, Frank’s formula
holds [38]:
b
b
≈ .
(1)
d=
2 sin(θ/2)
θ
In contrast to d, the dislocation core radius rm does not
depend on misorientation angle θ . Now the experimentally
observed cross-over between a regime of strong dependence
of Jc (gb) on θ (5◦ < θ < 20◦ , typically) to the regime of
weak dependence (for θ above some critical value) could be
naturally interpreted [23, 39] as a geometrical effect of the
decreasing distance between the dislocations. In particular,
in the range of small tilt angles the grain boundary critical
current, scaled by its intragrain counterpart, should follow
(2)
owing to the almost linear reduction of the size of
superconducting (unstrained) aperture, d − 2rm . For
larger tilt angles the dislocation cores overlap and the
grain boundary critical current saturates, in this model,
at the low level of intrinsic Josephson weak links. A
pronounced sensitivity of Jc (gb) to small magnetic fields
in this range is also consistent with the concept of a weaklink boundary. There are a large number of reports of
structural and transport studies on flux-grown and thin film
bicrystals, covering both YBCO- [22–26, 28, 33, 39, 41–43]
and BSCCO- or TBCCO-based [44–50] systems, claiming
at least qualitative accordance with the model of strain
induced by dislocations, imposed to underlie the order
parameter suppression. On the quantitative side, it is,
however, important to note a broad variation in reported
results and related conclusions concerning both different
families and sample-to-sample variations. For example,
there are reports that some high-angle boundaries do not
exhibit weak-link behaviour [42, 51, 52] at all; the presence
of a significant amount of high-angle boundaries in melttextured samples permits high supracurrent even in 1.5 T
at 77 K [20, 21]. Also, the precise location of the strong
link–weak link cross-over misorientation angle is very
uncertain and varies inside a broad interval, 5◦ < θ < 20◦ ,
depending on sample family, boundary type and method
used for its formation. For example, [001] tilt boundaries
in BSCCO-(2212) thin films are weak linked [44, 45, 49]
for misorientations above 5◦ –10◦ while equivalent naturally
grown bicrystal boundaries [50], in spite of their reduced
irreversibility field and increased grain boundary resistance,
are more consistent with strong coupling. The BSCCO(2212) and YBCO [001] twist boundaries also reveal
different behaviours: while the former can have depressed
Tc without being weak linked [53] in the latter the
depressed Tc always leads to a weak-link character of
the boundary [42]. Also, there is a broad variation in
dislocation core radii rm which one can extract, by applying
equations (1) and (2), from different observations (e.g.
rm = 2.9b [39] and rm = 1.2b [53]). In most cases rm is
larger than the dislocation cores in traditional systems (rm =
b [38]), a feature which has not been entirely understood
yet [3]. The model of a grain boundary strain field
therefore contains some intrinsic complexity so the attempts
to interpret the transport and structural data consistently
should be combined with other complementary models
of grain boundaries. For example, the cases of strongly
linked high-angle boundaries (instead of being weakly
linked as in the vast majority of cases) could be interpreted
inside the coincidence sites scheme [11, 39]. It should be
noted, however, that there are, generally, no substantial
and unexceptional correlations between the coincidence
index and the transport character of a boundary [53].
One could therefore conclude that the problems of order
parameter suppression are obviously based on structural
features of the boundary but also that there are no rigorous
and straightforward models, at present, able to provide a
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M Prester
full understanding of transport features of the boundaries.
Perhaps the primary reason for that is, taking into account
the approximately 1 nm scale of coherence length, the
insufficient spatial resolution and compositional accuracy
that present-day microscopy techniques provide, especially
concerning the compositional and structural order of oxygen
in the boundary region.
The fact of a complex and/or heterogeneous boundary
structure, documented to exist ‘among as well along
the boundaries’ [3], stimulates the alternative, rather
phenomenological, approaches to the problem of order
parameter suppression or, generally, of transport across
the boundary. These approaches concentrate, relying
more on boundary disorder than on its spatial order, on
microbridges [54] or nanobridges [55, 56] which may be
formed, for a number of reasons, as localized supercurrent
links between the grains. The evidence for these links
comes primarily from considerations of transport features
of the boundaries, i.e. from the studies of the character of
Josephson coupling across the boundary in particular. Now
we briefly review this important subject.
3.2. Mechanisms of Josephson coupling across the
grain boundary
A weak interaction between the superconducting grains is
responsible for the well-documented Josephson behaviour
of an HTS boundary. Quite generally, this interaction may
be realized, depending on the electrical features (insulating,
normally conducting, or comprising local superconducting
shorts) assumed to characterize the boundary, by one of the
following processes:
(i) tunnelling of Cooper pairs across a forbidden
(electrically insulating) boundary interface, usually referred
to as an SIS scenario;
(ii) overlap of superconductive wave functions of
the grains in the normally conducting boundary, usually
referred to as an SNS, or proximity effect, scenario;
(iii) supercurrent transport through tiny (spatial scale
ξ ) superconducting constrictions (‘pinholes’) bridging the
grains, sometimes referred to as a point contact scenario.
Theoretical elaboration of each of these processes leads
to specific predictions concerning experimentally accessible
quantities of a boundary. The most useful one is the
critical current of a boundary and its dependence on
temperature. In all cases, (i)–(iii), the magnitude of the
critical current is inversely proportional to Rn , the normalstate resistance of a junction, predicting, however, different
temperature dependences as well as different dependences
on geometrical and electrical parameters of the junction
and abutting superconducting banks. Experimentally, the
value of Rn is given by the slope of the I –V characteristic
in its linear (high-current) range. It is important to
note that there is a slight difference between the physical
backgrounds behind Rn in the SIS and the SNS cases. In
the SIS case Rn reflects tunnelling of quasiparticles and
is determined by the product of the square of the density
of quasiparticle states at the Fermi energy with the matrix
element that describes the probability for tunnelling across
338
the insulating barrier [32, 36]. In this case Rn is therefore
basically temperature independent in a broad range of
low temperatures [32, 36]. In the SNS case Rn reflects
the normal dissipative processes in the normal metal and
some temperature dependence of Rn may be expected, at
least in a broad temperature range. In particular, in the
simplest case of the best-understood tunnelling junction,
the Ambegaokar–Baratoff [57] formula gives
eIc Rn =
tanh 1(T )
π
1(T )
2
2kB T
(3)
where 21 is the energy gap of the superconductor involved.
In the other two cases (i) and (ii) the results substantially
depend on the applicability of either the ‘clean’ or the
‘dirty’ limit [37], determined, in turn, by a relationship
between the length scales involved (mean free path l in
the normal layer and superconductive coherence length
in both superconductor and the normal layer, ξs and
ξn , respectively) and the geometrical parameters involved
(radius of bridging constrictions r and the interlayer length
L). The results for Ic Rn acquire simple forms in some
limited cases when the sinusoidal Josephson relationship
Is = Ic sin φ between Is (actual supercurrent) and φ (order
parameter phase difference) holds. Thus, for an SNS
junction in the dirty limit (l ξn ), characterized by a gap
magnitude at the S–N interlayer 1i (T ), and at temperatures
T close to Tc , one has [37]
eIc Rn =
π 12i (T ) L −L/ξn
e
.
2 2kB Tc ξn
(4)
The Ic Rn products associated with point contacts
(case (i)) were shown [58] to obey equation (3) (the
prefactor 0.5π being, however, replaced [35] by 0.66π ),
provided that the conditions on gap and constriction
sizes 1(T ) kB T , a ξs , are fulfilled [35, 58].
There are a large number of experimental reports on
HTS structures comprising artificial or natural boundaries–
interlayers claiming evidence in favour of each particular
type of weak link. An exhaustive reference list may be
found in the recent review by Delin and Kleinsasser [35]
which also shows that the assignment of a proper model to
particular experimental results is, generally, not a simple
task. In this work we primarily focus on the Josephson
character of grain boundary weak links. The experimental
aim of most of the investigations is the temperature
dependence of both the critical current Ic and the magnitude
of the Ic Rn product. Knowledge of these data would
allow comparison with predictions of specific models (e.g.
equations (3) and (4)) and consequently also a conclusion
on the weak-link interaction responsible.
3.2.1. Ic (T ). The temperature dependence of Ic of the
boundaries in most cases obeys a power law of the type
Ic ∝ (1 − T /Tc )n , the exponent n acquiring values usually
in the interval 1–2.5 [22, 26]. There is a pronounced
sample dependence in the experimental exponent values
as well as a significant dependence on the width of the
temperature range employed in the corresponding fit to
the power law. In various polycrystalline forms (bulk
Current transfer and initial dissipation in HTSs
samples, polycrystalline films) the linear Ic (T ) dependence
(n = 1) is experimentally very common. The concavedownward (n < 1) and concave-upward (n > 1) Ic –T
dependences [59, 60] have both been reported, reflecting
probably more the evidence of self-field effects than the
ordinary sample dependences [60].
3.2.2.
Rn and Ic Rn product. The investigations
of bicrystalline and naturally grown YBCO boundaries
revealed at least two ubiquitous features of Rn : its
basic temperature independence [22] in the whole
superconducting temperature range and a high effective
resistivity ρn one can associate with Rn [22, 33, 61]. A
typical value at 4.2 K is ρn = 0.1  cm or, in terms
of specific contact resistance ρc (resistance multiplied by
area), ρc = 10−8  cm2 . The resistivity is therefore almost
3 orders of magnitude higher than the in-plane resistivity of
YBCO just above Tc . The values of the Ic Rn product for
most samples were found to be between about 1 and 4 mV
at 4.2 K [22, 33] although the range of measured values is
2 orders of magnitude wide. The product is obviously not a
constant determined primarily by 1 and seems to scale with
Rn−1 on average [33, 55, 56, 61]. This result may serve as a
basis for additional interpretations of the transport features
of a boundary [33, 55, 56, 61].
As far as Ic (T ) dependences are concerned all
three models may recieve partial experimental support [22, 23, 34] as all of the predicted behaviours may resemble, given an appropriate choice of parameters and their
values, the experimental power laws. However, the SNS
model reaches this agreement only under the assumption of
physically doubtful values of fitting parameters [35] and in
a limited temperature range below Tc . There are two main
reasons why the SNS (proximity effect) scenario does not
seem decisive in interpretation of the supercurrent transport across the boundary. The first one is an absence of
the predicted, principally exponential Ic (L, T ) behaviour
(equation (4)) in a wide temperature range (or the range
of junction–boundary lengths L). The other reason is that
the high effective boundary resistance, as well as its basic
temperature independence, is inconsistent with the notion
of any normal metal interlayer (which would play the role
of ‘N’) so the additional insulator barriers would have to
be introduced at the two S–N interfaces [62]. The latter would transform, in turn, the originally assumed SNS
into a more realistic but also less tractable SINIS sandwich
structure [35] in which the decisive processes take place
at the insulator interface [62]. In any case it is clear that
these interfaces play an important and probably unavoidable role in understanding the supercurrent transport across
the boundary. Indeed, the SIS prediction for Ic (T ), equation (4), agrees at least equally well [26] with the experimental results, given the assumption that a reduced energy
gap (1 ≈ 5 meV) characterizes the vicinity of an otherwise homogeneous boundary. This follows directly from
equation (3) which claims that the Ic Rn product at low
temperatures should be close to the energy gap, i.e. to approximately 20 meV (assuming the BCS relationship [64]
21 = 3.5kB Tc ). (It should be noted that the experimental results for the Ic Rn product compatible with the SIS
prediction simultaneously support the point contact scenarios owing to the common expression, equation (3), which
applies both to SIS and to narrow superconductive constrictions.) In more detail, there are two complementary
approaches which are both in basic agreement with the SIS
model and the experimental results. The first one treats
a boundary as a homogeneous object, assuming a reduced
gap to characterize its vicinity [22, 24, 26]. The second one
treats a boundary rather as a disordered, inhomogeneous
object, comprising ohmic shunts in parallel with its localized SIS links [55, 56, 61].
In the homogeneous case there are several models
for order parameter reduction which may be applied
to the active boundary region. Quite generally, it is
plausible to assume that, because of the spatially monotonic
1 (a solution of the Ginzburg–Landau equations), the
gap function which vanishes at the insulating boundary
is also substantially reduced in its vicinity. Indeed,
de Gennes [37] calculated the analytical forms for an
intrinsic spatial variation of the order parameter near
the interface demonstrating the reduction. (The latter
strictly applies only to the S–N interface only but it is
certainly qualitatively correct for the S–I one as well.) The
carrier deficiency near the boundary, which can take place
for various reasons but primarily as a result of oxygen
miscomposition, could also be a cause for depression of 1.
Reduced experimental values of the Ic Rn product
could be also attributed to the effects of grain
boundary inhomogeneities. The latter applies in particular to those inhomogeneities which provide ohmic
(non-superconductive) shunting channels in parallel with
the localized, SIS-type weak links [33, 61]. As well
as a general and well-documented agreement with the
RSJ model, the existence of ohmic shunts receives
strong support from the scaling properties of experimental
junctions [33, 61, 65, 66]. Although a precise form of
scaling is difficult to determine a general trend for Ic Rn
to increase with Rn−1 (i.e. linearly with Jc or conductivity
σn ) is beyond any doubt. A phenomenological description
of such a boundary has been suggested by Russek
et al [33] and Moeckly et al [61], claiming that
the alternate de-oxygenated and properly oxygenated (but
still disordered) grain boundary segments introduce the
filamentary connections across the boundary. The local
oxygen order has been found to be rather unstable and
subject to reversible migrations [61]. The filamentary
connections across the boundary depend in essence on the
matching of segments belonging to the two abutting grains.
The distributions of segments on both sides of a boundary
have been found [61] to be not mutually correlated,
introducing a random distribution of ohmic and weak-linklike filaments along a boundary. This random distribution
favours [61], as well as disorder inside a boundary, the
scaling behaviour Ic Rn ∝ Rn−1 . The latter scaling has
also been recognized to fit well the microscopic analysis of
charge transport across the boundary by Halbritter [55, 56].
This approach focuses on resonant tunnelling as the process
most responsible for the phenomenology of grain boundary
transport and therefore deserves special attention.
339
M Prester
3.3. Tunnelling across the grain boundary
Among various classical transport mechanisms, tunnelling
of charge carriers has been claimed [55, 56, 67] as the
one which characterizes in essence the normal and the
superconducting transport across the boundary. Both the
classical tunnelling of quasiparticles through an insulating
grain boundary barrier and resonance tunnelling, a process
mediated by charged impurities of the grain boundary, have
been invoked and microscopically analysed [55, 56, 67]. A
principal reason for ranking the importance of tunnelling
processes so highly is primarily related to the specific
combination of intrinsic material parameters of actual HTS
systems which places them in the immediate vicinity of
the metal–insulator transition (MIT) in the corresponding
phase diagrams.
The magnetically ordered insulator
phase is known to be a generic parent phase of HTS
cuprates [5, 6]. This phase indeed stabilizes provided that
an appropriately low density of hole carriers is present
in the quasi-two-dimensional conduction band of [CuO2 ]
planes. A charge transfer (‘doping’) from the nearby charge
reservoirs [68] regulates the actual hole concentration and
the critical carrier density of the MIT has been found [69]
to be of the order of 1021 cm−3 . This unusually high
density, in comparison with the typical values of the
order of 1018 cm−3 for three-dimensional systems [70],
can be naturally attributed to the effective two-dimensional
conduction in HTS cuprates.
Now applying these
circumstances to the problem of charge transport across the
HTS grain boundary one finds that the orders of magnitude
higher resistivity of the grain boundaries, compared with the
in-plane intragrain resistivity, means that the boundary is
far in the insulator side of the MIT. A dominant mechanism
of electrical transport across such an interface is tunnelling,
which should equally apply both to the normal and to the
superconductive state of the abutting intragrain compound.
Instead of mobile carriers the boundary contains a high
concentration of charged localized sites (of the order of
1021 cm−3 ) and their presence substantially influences the
conductive properties of a boundary. As well as the
conventional ‘direct’ quasiparticle tunnelling across the
insulating interface [32, 36] the charged impurities give rise
to an additional, impurity-mediated channel of electrical
conduction, i.e. to resonant tunnelling. The current of
Cooper pairs is also influenced by the presence of localized
sites due to on-site Coulomb interaction. This repulsive
interaction locally counteracts the superconducting state,
e.g. by inducing pair weakening [71], so the order parameter
becomes a complicated spatially varying function along
the boundary. The experimental macroscopic quantities
of a boundary, Rn and Ic , stem therefore from the local,
tunnelling-site-related variables [55, 56] jci , Rni and 1i :
X
jci Ai
(5a)
Ic =
i
X 1
1
=
Rn
Rni
i
eIc Rni =
340
π
tanh 1i (T )
1i (T )
2
2kB T
(5b)
(5c)
where some averages inside the small grain boundary
area Ai are assumed. Taking into account all relevant
transport mechanisms (both dissipative and non-dissipative
ones) the microscopic treatment predicts the validity of
the scaling law jc Rn ∝ Rn−1 , in full agreement with the
experimentally established conjecture [33, 61] mentioned
above. A previously introduced notion of ‘parallel ohmic
shunts’ receives therefore not only support from the model
of resonant tunnelling but also a reasonable microscopic
foundation.
4. Current transfer in high-Tc superconductors:
global aspects
Various forms of macroscopic HTS samples (excluding
perhaps the perfect single crystals and epitaxial films)
contain generally a large number of grain boundary weak
links. The global transport properties of macroscopic
samples, focused on by this review, certainly depend on
the properties of the grain boundary ‘building blocks’,
as they were summarized above, but also on other
intragranular (intrinsic) and microstructural features of
the HTS sample under consideration. In particular, the
problem of dissipative and non-dissipative currents in such
a medium includes, for example, a microscopic physics
of intragranular transport phenomena which are not fully
understood yet. The latter applies both to transport in the
normal phase [6] (e.g. to the temperature dependence of
resistivity) as well as in the superconducting one [6] (e.g.
to mechanism and symmetry of pairing). However, if one
considers the problem of charge transfer on a spatially
macroscopic scale, i.e. on the experimental scale which
is coarse enough to allow averaging out of the subtle
intragranular transport features (e.g. multiple branching
of a single crystal’s I –V characteristics due to intrinsic
interplane Josephson junctions along the c-axis [72]), the
charge transport is indeed primarily determined by the
electromagnetic properties of individual grain boundary
weak links, their spatial distribution and the statistical
distribution of their properties. A pronounced role of
grain boundaries in charge transport phenomena is simply a
consequence of the high effective resistivity of a boundary,
compared with the resistivity of the volumetrically
predominant and, on averaging, homogeneous intragranular
background. This qualitative relationship between the
intragranular and grain boundary resistivities, documented
to exist both in YBCO and in BSCCO systems [50, 73, 75]
has been pointed out in numerous reports as a plausible
source of the inhomogeneous distribution of normal and
superconducting currents. In particular, it has frequently
been suggested that the actual current lines bypass all
high-angle boundaries, or boundaries substantially degraded
by other causes, owing to their high resistances (decisive
for normal transport) or because of their small Josephson
critical currents (decisive for supercurrent transport). The
macroscopic currents are supposed therefore to meander
(‘percolate’) around the resistive obstacles, minimizing the
overall dissipation, very similar to the general concepts
of transport in classical percolative systems [76]. There
are various fundamental and applicative reasons why this
Current transfer and initial dissipation in HTSs
concept of microstructure-induced inhomogeneous current
transport attracted, in a variety of approaches, a lot
of attention. The intriguing points are, for example,
the obvious links with general ideas of transport in
heterogeneous media, the formulation of analytical models
for macroscopic transport which would take into account
the specific weak-link properties of a boundary and the
current capacity improvements that the promising HTS
forms (Ag-clad tapes, thin films) may recieve from detailed
and now even visually explicit knowledge of current
distribution in these samples etc.
In the rest of the paper we review these different
standpoints by grouping them into three main categories.
The charge transport can be treated as a problem (or a
subject) of
(i) percolation in electrically heterogeneous networks,
(ii) a Josephson-coupled medium with microstructuredependent parameters (brick-wall model, railway-switch
model)
(iii) spatial distribution of local magnetic induction and
supercurrent lines in real samples (magneto-optical and
scanning Hall probe studies).
Of course, many different approaches belonging to
these categories share similar ideas so the classification
introduced above is meant to reflect rather the ‘pedagogical’ aspects of elaboration of the problem of charge
transport in HTSs, not to systematize possibly confronted
standpoints about mechanisms which underlie experimental
observations. For example, one of the common ideas is the
aforementioned percolation that pertains, albeit not entirely
within the same context, to most of the approaches classified in categories (i)–(iii). The first group of approaches
deals explicitly with this specific transport phenomenon.
4.1. Percolation in electrically heterogeneous networks:
disordered-bonds model
A pronounced order parameter suppression, taking place
at grain boundaries in HTSs, justifies the interpretation
of transport problems in polycrystalline HTSs inside the
model of granular superconductors [77]. Traditionally,
the latter term applies to macroscopic assemblies of
low-temperature superconducting grains (or other types
of uniform superconducting islands) able to maintain
macroscopic phase coherence under the assumption of the
appropriate strength of Josephson coupling at the grain (or
island) interfaces [78]. The free energy expansion (F ) of
such a system is [77, 79]
X
X
2
4
2
F =
|9i − 9j |
Vi (a|9i | + b|9i | ) + c
(6)
i
j
where 9i,j are Ginzburg–Landau order parameters (small
by assumption) associated with the grains i and j of
volumes Vi,j and a, b are the usual Ginzburg–Landau
coefficients.
The last term defines the intergranular
Josephson coupling.
This formula is a clue for understanding various
transport features of granular (low- [79] and high- [77, 80]
Tc ) superconductors once the applied magnetic field and
intergranular coupling are known. In the case of strong
intergranular coupling, measured in units of condensation
energy per grain, the theory of an inhomogeneous system
reduces to the theory of homogeneous superconductors in
its dirty limit. The case of HTSs corresponds, however,
to weak coupling [79] and the heterogeneous structure has
to be taken explicitly into account [77, 80]. In general,
one has to consider vortices and their dynamics which
play a decisive role in studies of current distribution
and the onset of dissipation. However, in cases when
the presence of vortices may be disregarded (as in the
absence or very small magnitude of an applied magnetic
field), which we primarily focus on in this review, the
problem of related charge transport may be formulated, as
elaborated in a number of papers, inside a quite general
framework of conduction in a heterogeneous medium. This
phenomenological approach extends in part the related
work on low-Tc inhomogeneous superconductors [78] and
superconductor–normal metal composites [81].
If the model of a heterogeneous medium is to be applied
to the problem of charge transfer in polycrystalline highTc superconductors one has to identify first its specific
components, i.e. the subsystems differing substantially
in their conductivities (or, generally, current capacities).
Quite generally, the two mechanisms of conduction
involved in the charge transport in HTSs discussed so
far, i.e. quasiparticle or Cooper pair transport inside the
intragranular two-dimensional conduction band and the
tunnelling of charge carriers across the interfaces, provide
a natural microscopic background for heterogeneous
conduction. As these mechanisms are localized inside
either intragranular or grain boundary regions it is generally
agreed that one of the required subsystems comprises
isolated superconducting grains while the other comprises
grain boundary interconnections (Josephson junctions).
The latter subsystem, usually called a weak-link network
(WLN), is expected to play a central role in macroscopic
current transport. Experimentally, a number of results
of transport measurements identify contributions belonging
to each of the subsystems, or directly prove the reality
of the WLN. In particular, a.c. susceptibility results may
be consistently interpreted as an evidence of growth of
global superconductivity as a two-stage process [83]: first,
a local (intragranular) superconductivity takes place and
then, at lower temperatures, a global phase coherence [84]
sets in. Accordingly, the two maxima in the imaginary
part of the a.c. susceptibility signal can be directly related
either to intragranular or to sample-sized (WLN-mediated)
supercurrent loops. Even more direct evidence of WLNs,
as a network in the conventional sense, may be drawn
from combined current–voltage (with temperature as a
parameter) and temperature–resistance (with measuring
current as a parameter) characteristics of polycrystalline
samples in the YBCO family [85].
The resistive
transition successively measured with measuring currents
inside a broad interval (5 orders of magnitude) reveals a
pronounced branching in the lower part of the transition
while the upper part remains identical for all currents
(figure 3). A well-defined position of the branching
341
M Prester
1.0
3
T=80K
H=0 Oe
2
1
dV/dI (mΩ)
normalized resistance
2.9 mΩ
0.5
0
0
2
4
current(A)
6
2.9 mΩ
$
µ$
0.0
80
84
88
92
96
temperature(K)
Figure 3. Resistive transition of a GdBa2 Cu3 O7−x sample,
subsequently measured with the measuring current in a
broad range (5 orders of magnitude) [85]. A well-defined
branching point separates the ohmic region, where all of
the curves overlap, and the shaded rectangular non-ohmic
region. The sample resistance in the branching point
coincides with the quasi-ohmic saturation of dV /dI curves
(inset). The position of the branching point systematically
depends [85] only on the microstructure (i.e. on average
grain size) of the samples. Non-ohmicity stems from the
temperature- and current-dependent number of excited
(dissipative) grain boundary weak links, limited from above
by their total number. The branching point corresponds
therefore to all available boundaries in a dissipative state
while a predominant fraction of the sample volume (grain
interiors) is still non-dissipative.
point in the temperature–resistance diagram systematically
depends only on the sample’s microstructure (i.e. on
average grain size). The value of the resistance in that
point coincides with the almost temperature-independent
quasi-ohmic saturation of I –V characteristics [59, 86, 87].
These measurements demonstrate a complete separation of
dissipative excitation, in the absence of a magnetic field,
between the WLN sites (i.e. grain boundaries) and the
intragranular background [88] (figure 3). In that case the
localized dissipation pertains only to a discrete set of WLN
nodes, with their total number limited by sample size and
microstructure. The actual number of dissipative sites,
i.e. the dissipative fraction p, depends, up to the value
p = 1 reached in the branching point or in the quasi-ohmic
saturation of I –V curves, on the applied current and actual
temperature [85, 89]. Macroscopic charge conduction in
WLN of high-Tc superconductors reflects therefore an
interplay between local and global processes.
A clear correspondence which can obviously be
established between WLNs and classical heterogeneous
networks stimulates the approaches which interpret the
charge transport in WLN-limited HTSs as a rather
general problem of heterogeneous media. In these media
the global transport features are determined, assuming
appropriate disorder of the network under consideration,
by principles of percolation theory [76]. A problem of
conduction in random, electrically conductive networks,
such as random-resistor networks (RRNs) or randomsuperconductor networks [76], represents a well-known
example of the latter theory. The internal composition of
342
Figure 4. Disordered-bonds model shown schematically.
Phase-coherent grains (disordered rectangles) are
interconnected by junctions in one of the two possible
states, i.e. with supercurrent on or off, determined by local
conditions of current density, magnetic field and
temperature. Supercurrent paths are represented by
meandering lines.
these networks is subsequently and monotonically varied in
such a way that the fraction p of (super)conducting sites
(or bonds) is replaced at random with the isolating ones.
A global charge conduction in a macroscopic sample exists
only for p > pc , where pc is a characteristic (percolation)
threshold. Close to pc all macroscopic observables vary
as a power law function (p − pc )n . The resistance of
RRNs disappears in particular as (p − pc )t , the exponent
t acquiring values t ≈ 2 (three-dimensional RRNs) or
t ≈ 1.27 (two-dimensional RRNs).
In heavily disordered WLNs of real HTS samples it
may be expected that percolation plays an important role
as well, controlling the interplay between the local current
and a global phase coherence. The conductive status of
WLNs obviously depends on the applied current itself
and a simple disordered-bonds model [88] (figure 4), may
be assumed to underlie the transfer of supercurrents and
the non-ohmicity of WLNs. In particular, a number of
experimental observables and/or transport phenomena of
HTSs were brought, either theoretically or experimentally,
into the context of percolation theory.
The most
relevant and the most widely discussed phenomena are
critical currents, initial dissipation and current–voltage
characteristics, penetration depth, resistive transition,
metal–insulator transition and resistance noise. Now we
review the representative results.
4.1.1. Critical currents. In a somewhat simplified
concept of critical currents in superconductors one could
propose that any superconductor biased with increasing
applied current reveals the two characteristic ranges: for
small currents there is no dissipation while for large
increasing currents dissipation rapidly (either linearly or
non-linearly) develops. The current which separates these
two current ranges is called a critical current irrespective
of any particular underlying mechanism. At temperatures
different from 0 K the latter scenario is oversimplified
owing to the presence of fluctuative residual dissipation
for any current. In a fluxoid (or vortex) lattice, for
example, the critical current is associated with depinning
of vortices which may participate, as thermally activated
Current transfer and initial dissipation in HTSs
events, in residual dissipation at quite low values of applied
currents. In the case of HTSs [90], particularly because
of the enhanced role of thermal excitations, depinning
does not necessarily involve any threshold-like current
and the term of the characteristic current seems (e.g.
in vortex liquid or vortex gas phases) to be a more
appropriate one. Actually, in the latter case there are
several characteristic current scales which determine a
particular dissipative range [90, 91]. The phenomenology
of vortices, being the subject of thorough reviews and
reports [90–92], will be briefly elaborated (however, only in
those aspects which seem relevant to us for WLNs of HTSs)
in section 4.1.2. Here we mainly concentrate on critical
currents which may be directly related to heterogeneity and
disorder in HTSs (i.e. to WLNs). Considering a disordered
WLN the corresponding supercurrents are constrained to
favourable (phase-coherent) network paths and are subject
to spatial branching on all scales above the mesoscopic
scale of average grain size. The critical current in such
a medium indeed separates the range containing samplesized supercurrent paths from the one compatible with only
smaller-scale paths. The simplest case of a weakly coupled
granular system composed of grains of uniform size a0
and Josephson critical current I0 would have a macroscopic
critical current density [86] I0 /a02 above which the phase
coherence disappears. However, disorder of both grain
sizes and local intergranular Josephson currents, together
with the uncertain effective value of a0 , as discussed above,
are the reasons why this result is of very limited validity: a
realistic model should take the effects of disorder explicitly,
i.e. ab initio, into account. Various existing models [90, 93]
do that by considering two-dimensional networks composed
of ideal Josephson devices at the networks’ sites or bonds.
The disorder enters the problem in a way which is specific
for particular model. In some models [93] randomly
distributed fractions of superconducting bonds and ohmic
resistors are assumed while the others [94] introduce, in
order to reflect the random orientation of grains, random
coupling strengths between neighbouring grains (network
sites) obeying simultaneously the parametrization based
on experimental data of Dimos et al [22]. In both
models the conditions of a current-controlled experiment
were assumed, analysing the current distribution and the
magnitude of a current above which there would be no
longer any supercurrent path along the sample and when,
for the first time, there appears a voltage. The latter current
is known as a critical one. In order to obtain it Leath and
Tang [93] started with the Ginzburg–Landau equation and
the Kirchhoff rule (current conservation) at each network
node, taking into account also the defects of various
types [95]. A correspondence to breakdown phenomena
in several randomly disordered systems has been firmly
established, mainly through the common presence of the
most critical defect or bond [95]. A power law current–
voltage dependence, V /L ∝ (Iappl − Ic )x , where L is
the linear size of the network, has been suggested by
numerical modelling. It has also been predicted that the
critical current density, Ic /L, vanishes logarithmically in
the thermodynamic limit (L → ∞) while the exponent x
approaches 3.0 in two dimensions. Rhyner and Blatter [94]
calculated, on other hand, the critical current by finding
the critical path, i.e. the interface which minimizes the
sum of local intergrain Josephson critical currents. For
a given disorder this path can be exactly and uniquely
determined so the critical current is simply given,
once
P
the critical path is known, as the sum Ic =
l∈path icl .
The currents higher than Ic produce a voltage along the
current direction (i.e. dissipation) since the critical path runs
across the sample, between the current feeding contacts. In
contrast to the prediction of report cited previously [93], the
critical current density has been found to be constant in the
thermodynamic limit and represents therefore a meaningful
quantity characterizing the network.
The supercurrent distribution obeys in these approaches
either the local charge conservation (Kirchhoff’s rules) [93]
or the scheme of linear optimization [94]. In both cases the
predicted non-uniform current distribution complies with
the usual models of percolation theory. There are also
several other reports which directly relate the percolation
threshold concentration, inside a disordered-bond model
(figure 4) of geometrical connectivity [85, 88, 89, 96], to
the effective critical currents. In low applied currents and
at low enough temperatures an HTS sample is multiply
connected by supercurrent paths (or by a ramified phasecoherent cluster [88]) owing to a predominant fraction of
undercritical junctions. With increasing current this fraction
continuously decreases following generally a non-linear
functional dependence p = p(I ) [89]. The probability
of finding a sample-sized supercurrent path disappears
at p = pc . In other words, these reports define the
critical current Ic simply as a current which satisfies the
relationship pc = p(Ic ).
4.1.2. Initial dissipation and current–voltage characteristics. There are numerous papers published so far which
report the current–voltage characteristics of polycrystalline
bulk and thin film samples. Results obtained in the absence of or in small magnetic fields, the conditions we
are primarily interested in this review, are, however, less
abundant owing to the technically demanding circumstances
of such measurements (large Joule heating at contacts by
high measuring currents, a detrimental effect especially in
thin films at low temperatures). In disordered WLNs of
HTSs the increasing applied current induces dissipation due
to normal conduction first in grain boundaries, as directly
demonstrated by locating precisely the ‘hot spots’ in spatially resolved resistivity measurements [97]. These localized excitations may be assumed to play a decisive role
in analytic forms of experimental I –V characteristics of a
disordered network as well. However, even in the absence
of an applied magnetic field the vortex dynamics could be
responsible, in principle, for measurable dissipation. The
vortices may originate from the self-field of the measuring current or the trapped environmental field or may be
introduced as topological excitations, i.e. as free or bound
vortex–antivortex pairs or thermal fluctuations in the form
of circular vortex loops of various diameters. The I –V
results published so far claimed consistency with several
dissipative mechanisms, invoking either vortices or localized ohmic excitations. Figure 5 shows the first derivative of I –V characteristics (which can be measured with
343
M Prester
344
I<I
{ V=0,
V~(I-I ) , I>I
30
c
n
c
20
c
10
a)
0
0.0
0.2
0.4
0.6
10
1
differential resistance dV/dI (µ
µΩ)
substantially better voltage resolution) of Ag/BSCCO tape.
The experimental data are shown on graphs with various
axes in order to illustrate the level of compatibility with
several models reviewed in this section. First we briefly
outline the results which involve the dynamics of vortices
and then those results which are consistent with localized
excitations.
In the case of a very broad voltage window the
experiments on several HTS systems [98] favour the
Ambegaokar–Halperin model for a single Josephson
junction [99]. In this model the dissipation arises from
thermally activated phase difference slippages which, if
taking place periodically in time with the rate θ(t), result
in a d.c. voltage V = (h̄/2e) dθ/dt. The model has been
suggested to apply to weak-link networks as well [100].
In the range of small applied currents the analytic forms
of predicted I –V dependences are similar to Anderson–
Kim’s flux creep basic relationship (V ∝ eI ) which is easy
to understand as both processes are thermally activated in
nature. However, numerous reports show, starting perhaps
with the one on YBCO sinterates [101], that in the ranges
of low dissipation and in the absence of a magnetic field
the I –V characteristics are experimentally much better
described by power laws of the type V ∝ I a(T ) . The latter
form may have various physical backgrounds. In classical
superconductors this form was related [102] to the spatial
distribution of critical currents which inevitably exist in real
samples as a result of disorder and spatial heterogeneity of
the pinning force. This idea could equally be applied to
polycrystalline HTS [103, 104]. The power law form of the
I –V characteristics is also expected below the temperature
of the Kosterlitz–Thouless (TKT ) phase transition [105].
The elementary excitations in the absence of applied
current and field, both in two-dimensional homogeneous
superconductors [78] and in weak-link superconducting
arrays [106], are bound vortex–antivortex pairs. The system
of pairs is characterized by quasi-long-range order below
the phase transition temperature while above it free vortices
become more and more dominant. The applied current in
the low-temperature phase would exert a Lorentz force on
pairs tending to break them apart [107] and the resulting
I –V characteristic should reveal a power law form [108],
with the exponent value expected to jump from 1 (T >
TKT ) to 3 (T < TKT ). The phenomenon of the Kosterlitz–
Thouless transition is intrinsically two dimensional in
nature and its application to three-dimensional systems,
including films which are more than a few monolayers
thick, is rather demanding theoretically. However, the
power law form has been identified in the range of low
dissipation of I –V characteristics in all available forms
of various HTS systems and interpreted, in numerous
reports published so far [107, 109], as an evidence of
bound vortex pairs. It should also be mentioned that
there are experimental findings which are not compatible
with traditional understanding of the phenomenon of
the Kosterlitz–Thouless transition in superconductors.
For example, a recent report of high-precision I –V
measurements on a monolayer-thick film in zero applied
field [110] concludes (from the ohmic behaviour of initial
dissipation) that unbound vortices are present well below
the nominal Kosterlitz–Thouless temperature.
b)
0.1
0.0
0.2
0.4
0.6
10
1
c)
0.1
0.2
0.3 0.4 0.5 0.60.7
current (A)
10
1
d)
0.1
0.01
0.1
reduced current I-Ic (A)
Figure 5. (a ) High-resolution differential resistance
(dV /dI –I ) data of BSCCO-(2223)/Ag tape. The resolution
is limited by noise at the level of about 1 µ (equivalent to
a voltage resolution of about 1 nV). The full line in (a )
corresponds to a breakdown form (see text) with exponent
value n = 2. (b ), (c ) The same data on plots with one or
both axes logarithmic. The straight lines demonstrate in (b )
compatibility with V ∼ eI and in (c ) compatibility with
V ∼ I a . (d ) Logarithmic plot of the data using the reduced
current, i = I − Ic . The slope of the straight line
corresponds to (I − Ic )n , n = 2.
Current transfer and initial dissipation in HTSs
The power law form V ∝ I a can be also consistent
with dissipation associated with regular three-dimensional
vortices in some special situations. In the absence of
an applied field the vortices which could be assumed
here originate from the trapped or self-field or, more
importantly, can represent the fluctuations in the form
of vortex loops of various circumferences which may be
thermally excited in the Meissner phase [91, 110]. Given
the vortices, the power law form can be extracted from
the generalized Anderson–Kim flux-creep form [112] (V ∝
exp(−U/kB T )F (I /Ic ), where U is the vortex pinning
potential in absence of a current. This form allows
various types of apparent pinning potential, in addition
to the linear, Anderson–Kim one, F (I /Ic ) = 1 − I /Ic .
In particular, the I –V power law is compatible [113]
with F (I /Ic ) = ln(I /Ic ), the form suggested by Zeldov
et al [114]. There are several experimental papers which,
favouring I –V power law [74, 115, 116], follow the latter
interpretation. Alternatively, in the framework of the
weak collective pinning theory [90] or the vortex phase
transition concept [91, 117] the I –V power law corresponds
to the restricted (critical) temperature range around Tg ,
the temperature of the vortex liquid–vortex glass phase
transition [91]. Below Tg the vortices are long-range and
above Tg only short-range ordered. The order itself is of
the spin-glass type rather than of the hexatic lattice type of
classical superconductors. Therefore the isothermal I –V
characteristics belong to one of the two generic classes
associated with each of the phases and are characterized
by specific scaling properties. Around Tg the curves are
expected to be of the simple power law form V ∝ I a ) with
the exponent a related only to universal critical exponents
of the theory of critical phenomena. The experimental
support for the phase transition scenario can be found in
measurements on thin films and single crystals of HTSs
in magnetic fields of various intensities [117] but also on
polycrystalline samples in the absence of or in a small
applied field [118].
Now we review those experiments and related
mechanisms which claim consistency with the form V ∝
(I − Ic )a (hereafter called the breakdown form), instead
of with the form V ∝ I a just discussed above. The
breakdown form we discuss now introduces explicitly
the critical current Ic as a threshold current for the
onset of dissipation and is therefore closely related to
the notion of critical currents discussed in the previous
section. This form obviously neglects dissipative processes
which can exist in a superconductor, in principle, above
0 K at any current (or, perhaps, even at 0 K owing
to quantum tunnelling [119]). However, there are many
reasons which justify this form in the cases of both the
vortex dynamics and the localized normal excitations.
Concerning vortex dynamics in the absence of an applied
magnetic field, one could conclude, for example, that a
finite interlayer interaction in HTS samples influences the
originally two-dimensional (intralayer) Kosterlitz–Thouless
vortices and alters the effective vortex–antivortex coupling
both in real three-dimensional planar structures [120]
and in Josephson junction arrays [121]. The additional
interactions were shown to be responsible for non-vanishing
critical currents Ic . Consequently, the I –V characteristics
obey, instead of the traditional KT form V ∝ I a ,
the form [120, 121] V ∝ I (I − Ic )a−1 . In granular
superconductors one expects [78] a pronounced role of
localized quasiparticle excitations. The breakdown form
we discuss naturally arises in the context of a problem
of coherence [80] (or phase locking [78]) transition. The
important ingredient of the latter approach is a division of
the total applied current into two additive components, i.e.
into supercurrent and dissipative quasiparticle current. The
bulk critical current enters the problem as a decoupling
current, built up from the maximum Josephson current
of individual junctions. Theoretically, the problem of
Josephson-coupled grains is isomorphic with the X–Y
model of coupled spins so the low-temperature phase
coherent state can be thermodynamically stable against
fluctuation in three-dimensional systems, quite analogous
to the low-temperature stability of the ferromagnetic phase
in appropriate magnetic samples. The response to an
applied electric field of the three-dimensional Josephson
network can be calculated, a power law V ∝ (I − Ic )a
being obtained in a phase-coherent state above the
critical current [80, 122]. The experimental support was
originally provided by measurements on specially prepared
classical superconducting arrays [80] but soon afterwards
also by a number of papers reporting results on HTS
samples [82, 96, 123–126]. The interpretation of results on
polycrystalline HTSs includes usually other peculiarities
of these systems, e.g. the features related to glassy
behaviour [122]. In any case, the experiments justify the
concept of an almost dissipation-free region (I < Ic ) of the
I –V characteristics in the absence of or in a small applied
field. The latter region of transport I –V measurements
could be related to the regime of volume persistent
currents in magnetic measurements (in particular those
characterized by a slow temporal decay) well documented
to exist [127] in polycrystalline HTS rings. The onset
of dissipation (I > Ic ) in WLNs of HTSs, as measured
by I –V methods (including those employing high-voltage
resolution [88] of the order of 1 nV) involves therefore,
in this approach, primarily the localized quasiparticle
excitations. The reported experimental values of the
power law exponent and the interpretations concerning the
particular mechanism that the exponent expresses differ
somewhat from author to author. The work of Lebeau
et al [80] postulates an analogy between the supercurrentrelated excess conductivity and the susceptibility in the X–
Y model. At the coherence temperature Tcoh the exponent
was concluded to be a(Tcoh ) = 1 + γ /φ or, employing the
hyperscaling relationship [76] between critical exponents,
a(Tcoh ) = (d + 1)/(d − 1 + η). The exponents γ and
η correspond to critical behaviours of susceptibility and
correlation length, respectively, while φ is the cross-over
exponent [76]. The latter expression for a relates the
exponent exclusively to static exponents of the network.
For three-dimensional systems this expression predicts
therefore a(Tcoh ) ≈ 2 while at lower temperatures a
is expected [122] to increase linearly with decreasing
temperature. These predictions have been claimed to be in
good general agreement with several experimental reports
345
M Prester
346
10-5
10-6
H=0 Oe
90K
66K
10-7
voltage (V)
on classical granular [80] and high-Tc polycrystalline
[82, 123] superconductors. The experimental value and
the dependence of a on temperature and magnetic field
seem, however, to depend on the width of the voltage
range covered by the recorded I –V characteristics. A
standard I –V measurement usually covers, employing
the electric field resolution of 10 µV cm−1 , several
voltage decades. On the other hand, high-resolution
I –V measurements [88, 96, 124, 126] have focused the
very onset of dissipation inside a narrower window (the
upper field limit being typically 10 µV cm−1 ). These
results have also been found to be perfectly compatible
with the breakdown power law form, although with
somewhat higher and less temperature and magnetic field
dependent average values of the exponent a.
Nonsystematic [124, 126] and very weak [96] dependences on
temperature and magnetic field have both been reported.
The percolation (disordered-bonds) model for the onset of
dissipation has been provided by Prester [88], together with
a quantitative analysis of experimental results on various
polycrystalline HTS samples. The model is schematically
represented by figure 4. It has been argued [88], that
the I –V characteristic, taken at fixed temperature and
magnetic field, can be quantitatively modelled as a currentinduced percolation transition, characterized by cluster
dynamics known in classical (electrically conducting)
random networks. A driving variable of this transition is
p, the dissipative (or non-superconducting) fraction of the
network’s bonds which depends on current, temperature and
magnetic field; the latter two experimental variables are,
however, kept fixed and the remaining p(I ) dependence is
linearized in the model. The model proposes that in nonohmic WLNs the differential resistance (dV /dI ) replaces
the resistance (V /I ) of classical (ohmic) random networks;
the quantity V /I , a well-defined macroscopic transport
property at a given p in ohmic networks, loses its meaning
in non-ohmic WLNs. The dynamical exponents of random
networks, such as the conductivity exponent t or breakdown
exponent [76] s, have been proposed and experimentally
documented to play a major role in the onset of dissipation
and subsequent cross-over behaviour. In particular, the
exponent a (interpreted as t + 1) was found to be a ≈ 3 in
a rather broad range of temperatures and (weak) magnetic
fields and in three-dimensional samples much bigger than
their average grain sizes [88].
Figure 6 shows the
measurements of I –V characteristics in a broad temperature
range on a BSCCO polycrystalline sample. In figure 6(a)
the measured data are plotted on logarithmic axes, a
common procedure for a demonstration of compatibility
with a current–voltage power law [117]. The same axes
were used for plotting figure 6(b) as well; note, however,
that a reduced current i (i ≡ I − Ic ) was introduced instead
of I . The data were consequently transformed into a set of
parallel lines, indicating that the same physical mechanism
characterizes the onset of dissipation at all investigated
temperatures. The slope of the lines is a = 2.94 ± 0.04.
This value agrees well with t = 2, a widely accepted
value for the conductivity exponent in three-dimensional
random resistor networks [76]. In the disordered-bonds
model the power law form of the I –V characteristics
10-8
0.1
1
current (A)
10-4
10-5
10-6
10-7
10-8
10-2
reduced current i=I-Ic (A)
10-1
Figure 6. (a ) I –V characteristics of polycrystalline
BSCCO-(2212) in a broad temperature range (66 K–90 K)
and in the absence of an applied field. (b ) The same data
after the transformation to reduced current, i = I − Ic , has
been performed. The slope of approximately parallel lines
is 2.94 ± 0.04, related to the conductivity exponent (see
text) t ≈ 2 of percolation theory.
above Ic reflects therefore a scale invariance of macroscopic
properties associated with percolation networks close to pc .
This is completely analogous to scaling in ordinary phase
transitions in the temperature domain. The recent results of
dynamical simulations of I –V characteristics of Josephson
junction arrays [128] are in good quantitative agreement
with the predictions of the disordered-bond model.
4.1.3. Penetration depth. A common point of various
percolation models involving charge transfer in WLNs
of HTSs is the idea that macroscopic observables of the
network are principally determined by its composition,
e.g. by the fraction p of non-dissipative bonds. This
fraction determines the average size of phase-coherent
islands (clusters) so the deviation of p from pc , the latter
being the fraction associated with diverging cluster size,
is accompanied by a power law form of macroscopic
observables. This form reflects the scale invariance of
a particular observable and is characterized by a specific
exponent as well. The fraction p may be considered,
at least in some limited experimental interval, as a
function of the three independent variables I, H and
T . Macroscopically equivalent sites can be therefore
be achieved through independent action of each variable
while keeping the other two constant [88, 89]. As is well
known, a power law temperature dependence of various
experimental quantities studied in the theory of critical
phenomena may be interpreted, close to Tc (defined by
pc = p(Tc )), as evidence of spatially diverging coherence.
In the specific case of WLNs of HTSs the penetration
depth of thin YBCO films was shown [129] to obey a
Current transfer and initial dissipation in HTSs
power law temperature dependence in a broad temperature
interval. Theoretically, the percolation model predicts [130]
that the divergence of the penetration depth is determined
by the conductivity exponent t. Assuming the validity of
the specific model [129] for the unknown function p(T ),
the experimental data were found to be consistent with
theoretical predictions.
4.1.4. Resistive transition. Because of presence of
WLNs the resistive transition from the normally conducting
to the macroscopic superconducting state of HTSs is rather
complex. In polycrystalline HTS samples, in particular,
this transition is usually interpreted as a two-stage process:
the upper part corresponds to thermodynamic intragranular
transition while the lower part corresponds to intergranular
coherence transition. The former one may be described by
mean-field theory while the latter displays scaling properties
of critical phenomena [80]. The coherence transition
enables gradually the macroscopic (sample-sized) currents
and is also called, because of the slow approach to the
zero-resistance state, ‘a tail regime’ [101, 131]. The nonmonotonic features of the transition observed occasionally
in HTS samples [132] can be understood as a consequence
of competition between several dissipative contributions
characterized by various temperature dependences [132].
The current transfer in the tail regime has been both
qualitatively [131, 132] and quantitatively [80, 133, 134]
related to percolation. The analytical form of temperature
dependences was shown to be consistent with the power law
form. The relevance of either the three-dimensional X–Y
susceptibility critical exponent [80, 134] or conductivity
dynamical exponent [129] has been raised. The complex
and sample dependent exponent structure has been also
predicted [135] as shown by mapping the random
conductance problem into the problem of diffusion in
random-potential systems [135].
4.1.5. Metal-insulator transition. Transport properties
of HTS samples which differ in chemical composition
from the one optimal for superconductivity are very
intriguing in both single- and polycrystalline forms, as
is well known [6, 69], for many fundamental reasons.
Substitution of element(s) in the charge reservoir [68]
primarily induces a variation of charge carrier concentration
in the conducting layers and, consequently, also a drastic
degradation of electrical conductivity. A full range of
conduction types, from insulating to metallic behaviour,
has been experimentally demonstrated to occur in reality
(see, for example, [69]), reminiscent of the MIT in classical
systems [136] (e.g. doped semiconductors). For both
single-crystal and polycrystalline forms of HTS percolation
has been frequently emphasized as either an explicit
or an implicit framework for effective current transfer
mechanisms in both insulating and metallic phases; in
some reports [137] the MIT itself relies on percolation.
In particular, one can identify the hopping conduction
between the localized states as a responsible intrinsic
conduction mechanism in the insulating phase. Indeed,
there is a perfect accordance [69, 138] of experimental
results with the Mott–Davis[136] variable-range-hopping
formula for resistivity, ρ(T ) ∝ (T0 /T )α , where α = 1/4
in three-dimensional samples. As shown by Shklovskii
and Efros [139] there is a complete correspondence,
assuming participation of the two effective conductive
channels, between Mott–Davis resistivity and the resistivity
of critical percolating paths in a network of resistors.
The same quantitative temperature dependence has also
been obtained for percolative conduction in granular
systems, assuming tunnelling at grain boundaries [140].
At least one of these percolation models is therefore
effective in the interpretation of conduction in insulating
phases of bulk HTS materials. The compositions which
reveal metallic behaviour (i.e. which obey, at least in
a limited temperature interval at high temperatures, a
linear dependence ρ ∝ T , resembling scattering by Debye
phonons in ordinary metals) are either superconducting or
insulating at low temperatures. The conduction mechanism
in such ‘metallic’ samples may also involve percolation
of various origins [137, 138, 141]. Indeed, a very weak
saturation-type dependence of the superconducting critical
temperature on charge carrier density (instead depending
parabolically [142]), seems to indicate [137] the presence
of effective phase separation [143] into nominally metallic
(and below Tc superconducting) and dielectric components.
If this is the case the current transport could be governed
by percolation both above and below Tc . The percolation
can also be assumed to underlie the whole MIT. It should
be emphasized that the effects under consideration do not
necessarily involve only granular systems. The intrinsic
planar conduction in HTSs can be subject to quantum
percolation [138] which may dominate the transport in
system of localized but interacting carriers. The latter
system opens a Coulomb gap at the Fermi level [144] but
is simultaneously affected in HTSs by quantum fluctuation
due to the proximity of the superconducting state [138].
The hopping conduction in such a system is expected to
follow ρ ∝ T (‘metallic’ behaviour), obeying a single
mechanism of conduction in the whole range of the MIT.
In the latter case there would be no MIT in the traditional
sense but rather the cross-over from a non-correlated
localized system to a percolative system with a Coulomb
gap state [138].
4.1.6.
Resistance noise. Resistance noise of the
electrically conducting system may provide important
information on the conduction mechanism(s) involved in
electrical transport. Studies of the noise are especially
valuable when the system under consideration allows,
successively or simultaneously, more than one conductive
channel or source (such as the case of superconductors).
The studies of noise in HTSs have been reported
by numerous papers (for a recent reviews see, for
example, [145, 146]) which deal overwhelmingly with the
phenomena in the resistive transition. The experimental
quantity is the spectral power density Sv of the noise,
studied usually as a function of temperature. Apart from
a pronounced sample dependence in the details of Sv (T )
dependences, the studies of HTS films and sinterates
generally reveal a peak in the transition region, a clear
sign of the presence of different noise sources in the
347
M Prester
system [147]. Some of the results are obviously related
to noise generated by vortex dynamics in an applied
magnetic field [148] which is outside the scope of this
review. In the absence of a magnetic field the noise
source is closely related to structural disorder and defects
as shown by a number of studies revealing a proportionality
between defect concentration and the noise level (see, for
example, [146] and references therein). For example, the
presence of grain boundaries in polycrystalline HTS films
is considered to be a reason for the order of magnitude
higher noise level in these films compared with the noise
level of epitaxial films.
When the noise is related
to grain boundaries in polycrystalline samples it is still
necessary to answer the question of the nature of grain
boundary fluctuations as well as to model the integration
of their individual contributions into the resistance noise
signal measured on macroscopic samples. The model of
random switching [149] between the two possible grain
boundary states (normally dissipative and superconducting
non-dissipative ones) has been found to be more successful
than the traditional model involving thermal fluctuations
in grain boundary resistances. A decisive parameter of
the random switching model is the fluctuating number of
boundaries in one of the states. The same approach has also
been successfully used in studies of long-time relaxations
in current-biased ceramics [150] providing independent
support in favour of the model of random switching. On
the other hand, various percolation models which integrate
the individual switching contributions [145–147, 149] have
demonstrated their ability to describe quantitatively the
experimental noise results, particularly the Sv (T ) and Sv (R)
dependences. Although the effective scaling exponents
are not identical to those describing the noise in classical
random networks [149], the concept of percolative current
transport in heterogeneous networks can obviously be well
applied, as far as noise studies are concerned, to transport in
HTSs. The refinement of this model can be extended [146]
to include the other types of sample inhomogeneities which
are present even in highly oriented epitaxial HTS films.
For example, a spatial variation of Tc [146] can be related
to a local oxygen deficiency or other structural defects.
The resulting spatial current distribution can be studied by
low-temperature SEM [146], achieving a spatial resolution
of 1 µm, and revealing the correlation between the local
structural and transport features.
4.2. Microstructure-oriented models for macroscopic
current transfer: brick wall and railway switch
The previous sections emphasize mostly those aspects
of current transfer in HTSs which stem from several
elementary structural and electromagnetic properties of the
samples, i.e. from the very presence of a disordered network
of inter- (or intra-) granular weak links, charge conservation
and carrier conversion at the networks’ nodes, and from the
geometrical constraints imposed by the network itself. The
current transfer problem can also be approached starting
from the specific microstructure characterizing the real
samples (prepared usually by techniques which improve the
current capacity features of the sample, such as normalmetal/HTS tapes and thick films and well-oriented thin
348
films). Assuming a correlation between local structural
and electromagnetic properties, one can formulate a model
for (supra)current transport which would be able to predict
the charge transfer response and behaviour under various
experimental (and interesting in terms of applications)
circumstances. The first model of this type was a brickwall model by Bulaevskii et al [151]. The model
focuses in particular on the microstructure of BSCCO/Ag
tapes characterized by plate-like grains, which are rather
well aligned in the plane of the tape (i.e. in the rolling
direction). The tapes are produced by the process known
as the powder-in-tube technique. The crystallographic
c-axes of the grains are therefore preferentially oriented
normal to the plane of the tape. Although a real sample
may still include a number of grain boundary types it
would be reasonable to assume that the [001] twist and
[001] tilt boundaries dominate. The former type separates
the adjacent grains by plate faces and the latter type by
plate edges. The macroscopic (super)current transfer is
composed of components along the conducting a–b planes,
mutually parallel on both sides of the two boundaries
under consideration, but also of a component across one
or both of the boundaries. The main obstacle to current
transport can be attributed [151], especially in the presence
of a magnetic field, to small-area [001] tilt boundaries
and the model assumes that the intergranular current along
the c-axes dominates the macroscopic current transport,
even for current paths along the tape plane (figure 7(a)).
Given reasonable intragranular pinning, the Josephson
critical current density across the [001] twist boundary,
jc , was identified as a limiting factor of non-dissipative
bulk transport. In particular, if the microstructure is
modelled by a brick-wall comprising uniform rectangular
grains or bricks of sizes 2L, 2L, D the macroscopic critical
current density Jc should be given [151] simply by Jc =
jc L/D. In this model the critical current of a single twist
boundary determines therefore the global critical current as
well as its dependence on magnetic field and temperature.
Generalizing the model first to the case of disordered
boundaries [151] (with jc spatially dependent) and then to
the case of a microstructure comprising a distribution of
grain sizes and various boundary types [152] a variety of
experimental results can be interpreted, predicting also the
possibilities of improved critical currents. However, the
experimental results are not in unison regarding the direct
applicability of the brick-wall model to practical Ag-clad
tapes. While, for example, the presence of BSCCO-(2212)
phases at the [001] twist boundaries indeed influences [154]
the critical current of BSCCO-(2223)/Ag tape (favours the
model) the macroscopic current transport has been claimed
to involve little or no c-axis conduction [73] owing to
the absence of a specific temperature and magnetic field
dependence of Jc (opposes the model). Also, the presence
of an amorphous layer at [001] twist boundaries [155]
imposes a severe limitation on charge transport in the
c-direction. In agreement with this result, the anisotropy
ratio of the normal resistivity along the rolling plane has
been found [73] to be an order of magnitude smaller than
the single-crystal anisotropy ratio, indicating, in contrast to
the assumption of the brick-wall model, current transport
Current transfer and initial dissipation in HTSs
Figure 7. Microstructure-oriented models for current
transfer shown schematically. (a ) corresponds to the
brick-wall model (after [151, 152]) and (b ) to the
railway-switch model (after [153]). The thick lines illustrate
the percolative current transfer through the network of
strongly linked boundaries.
predominantly along the crystallographic a–b planes. A
sizable contribution to a–b conduction can obviously come
from [001] tilt boundaries as shown by an investigation
of naturally grown [001] tilt bicrystals [50]. If moderate
(θ < 8◦ ) misorientation of the boundary is present, the
results indicate elements of strong coupling between the
grains.
The model which relies on current transport inside
(001) (i.e. a–b) planes all along the tape is usually referred
to as the ‘railway-switch’ model [153], (figure 7(b)). In
addition to [001] tilt boundaries the transport along the tape
relies on components across the boundaries of various (or
mixed) structural types characterized by slightly misaligned
c-axes of adjacent grains. At the ‘switch’ the highly
conductive (001) planes meet generally at a non-zero
angle. The conductive status of a particular switch depends
on the misorientation angle of the neighbouring grains.
There are two structurally identified boundary types of
this sort which may be involved in conduction as ‘railway
switches’. The first is formed by misaligned grains stacked
one on top of the other, usually referred to as a c-axis
boundary [155] or (001) boundary [156]. The second and
perhaps a more important [155] boundary type is formed
by boundaries which involve no c-axis conduction, joining
the misaligned grains at the edges. They are usually
called a–b-axis [155] or (hk0) [156] boundaries. In all
cases, a small misorientation angle is necessary for a
‘switch’ to pass a substantial amount of current. A relative
abundance of small-angle boundaries would be therefore a
primary condition for the high current transport capacity of
a tape. Indeed, a detailed structural investigation of highquality BSCCO samples [157] revealed over 40% of all
boundaries to be small-angle boundaries, with an additional
8% belonging to coincidence-site lattices which may also
be strongly coupled irrespective of their misorientation
angles [51]. The large fraction of presumably strongly
linked boundaries suggests that supracurrents percolate
(figure 7(b)) through the network of enabled ‘railway
switches’ [157]. The percolative model of current flow
is also consistent with a sizable reduction of critical current
density of BSCCO/Ag tapes, compared with that of the best
epitaxial films [115] or the values ascribed to intragranular
transport [158].
As far as current paths in tapes
(produced by a powder-in-tube technique) are concerned
the recent reports seem to support overwhelmingly the
a–b conduction as described by various modifications
of the railway-switch model [73, 153, 155, 157, 159, 160].
Irrespective of the precise determination of current
paths the supracurrent-limiting mechanism can be rather
convincingly ascribed [161] to sample inhomogeneities of
various kinds [156] (including grain boundaries as a special
case), in particular in the absence of or in small magnetic
fields, given their ability to suppress the order parameter
in regions comparable in size with the superconducting
coherence length.
4.3. Visual inspection of percolative current paths
There is no doubt that the most convincing studies of
current transport in HTSs are those which provide visual
information on the current distribution in HTS samples.
There are variety of techniques developed so far: magnetooptical technique [162], scanning Hall probe [163],
scanning tunnelling [164] and SQUID [165] microscopy.
The results of other methods, developed primarily to
study localized magnetic structures (individual vortex or
vortex lattices), such as magnetic force microscopy, Bitter
decoration and electron holography are outside the scope
of this review. The visual data on the current distribution
in appropriate HTS samples, the subject of our primary
concern, are derived in all of these techniques from the
measured profiles of magnetic flux density; a calibrated
magnetic sensor maps the flux density and then the
appropriate algorithm determines the current distribution.
Although the current paths generated by application of
a magnetic field to an HTS sample are not necessarily
the same as the transport current paths [159], the current
distributions extracted from the flux density maps provide
perfect qualitative and quantitative information on samples’
electrical connectivity. Traditionally, the problem of the
distribution of supracurrents in a type II superconductor has
been described by the Bean model [1]. The model assumes
a long, homogeneous sample in a strong (H Hc1 )
magnetic field parallel to the long axis of the sample
and the conditions of quasistationary internal equilibrium.
In this case Ampère’s law (∇ × H = (4π/c)J ) has
simple solutions: there is either a constant magnetic-flux
gradient, accompanied by a spatially constant bulk current
density j = jc , or they are both zero (the current and the
gradient). The quasistationary equilibrium characterized
by a spatially constant bulk current density is known as
a critical state [1]. In this form the model, however,
applies only to the case of samples in the form of long
cylinders. A simple current–magnetic field relationship is
a consequence of the assumed geometry and of the absence
of demagnetizing fields. In cases of ‘flat’ geometries
349
M Prester
in a perpendicular magnetic field, which correspond to
conditions of magneto-optical and scanning microscopy
techniques for film and tapes, various specific geometries
can be treated both numerically and analytically. The
solutions significantly differ [166] from the Bean model
as discussed above. A common property is a pronounced
dependence of current–magnetic flux distribution on sample
geometry and shape. In particular, a rectangular sample
reveals geometrically regular domains characterized by
currents of uniform density and directions. The domains
are mutually separated by diagonal discontinuity lines (see
figure 8), where the current bends sharply, and along which
magnetic flux does not penetrate [167].
The latter analysis and models concern of course
homogeneous samples. The algorithm one should use
in order to obtain the current distribution in generally
inhomogeneous HTS samples has to be independent of
the assumptions of a critical state; a technique able
to measure and to map the current paths would be
highly desirable. Clearly, if it were possible to measure
a real three-dimensional magnetic induction B(r) with
reasonable spatial resolution the required J (r) could be
easily obtained by application of a few equations of
classical magnetostatics, i.e. ∇ × H(r) = (4π/c)J (r),
B = H + 4π M , ∇ × J (r) = 0, where M is the effective
magnetization associated with the induced supercurrent.
Although the spatial resolution of present-day sensors is
not a limiting parameter the three-dimensional maps of
B(r) would be technically rather demanding (if not almost
impossible) to obtain. Considering, for example, the case
of the flat-geometry samples, such as films and tapes with
the plane of the sample coinciding with the x–y plane,
the magnetic sensors measure only the z-component of the
magnetic induction Bz (x, y), which is generally insufficient
for determination of J (r). However, confining the currents
strictly to two dimensions, which is indeed a reasonable
approximation in a number of experimental situations (e.g.
in cases of small sample thickness on the scale of the
penetration depth), the Bz (x, y) data suffice for detailed
reconstruction of two-dimensional J (r). In particular, it
was shown by several authors (see, for example, [168]) that
knowledge of the magnetic induction profile above a planar
current distribution allows exact and unique reconstruction
of the current distribution itself. The algorithm inverts the
Biot–Savart law,
Z
r − r0
1
J (r 0 )
dr 0
(7)
B(r) =
c
|r − r 0 |3
which in its original integral form gives the induction
produced by the known volume current distribution J (r),
and calculates J (r) from the known (i.e. measured) maps
of B(r). Various inversion schemes have been proposed
so far [159, 169–171] and applied to HTS samples of
various geometries, including the samples of arbitrary
thickness [172]. The most widely used techniques for
magnetic induction mapping employ either miniature Hall
probes or magneto-optical coatings or films. A rapid
recent progress in the production of submicron Hall probes
enables measurements of magnetic induction with a high
spatial resolution (< 1 µm), keeping an acceptable level
350
of noise and intensity resolution [163]. Apart from
measurements of the magnetic profile of an individual
vortex [163] the scanning Hall probe technique has indeed
been used for reconstruction of the current distribution
in YBCO films [170] or BSCCO/Ag tapes [173], with
substantially lower spatial resolution of 25 µm or 100 µm,
respectively, being achieved, however. The majority of
results on macroscopic current distributions published so
far employ, however, various types of magneto-optical
imaging, the technique being recognized as a very powerful
tool for the investigation of a penetrated or trapped
magnetic field and local constraints on current transport
in superconductors (for a recent review see [162]). The
spatial resolution of magneto-optical techniques reaches
the submicron range [162], allowing, in addition to the
studies of current-limiting defects on the scale of 5–
10 µm, the observation of single vortices in low magnetic
fields [174].
4.3.1. Magneto-optical studies of current paths in highTc superconductors. Magneto-optical devices employ
the Faraday effect, illustrated in figure 8: the magnetic
field couples to the polarization (i.e. rotates the polarization
plane) of visible light travelling in magneto-optically active
materials [162]. Hence the films of these materials (e.g.
paramagnetic glasses, europium chalcogenides, yttrium–
iron garnets) enable, with standard polarization-resolved
optics, visual information on the spatial flux variation just
above the superconducting sample in the mixed state to
be obtained. In particular, the garnet (or bubble) films
were found [175] to be very convenient for studies on
HTSs, offering a spatial resolution of 5 µm typically
and a superior magnetic resolution [176] (up to 10 µT).
Magneto-optical imaging has been successfully used in
studies of flux penetration in all forms and families of HTS
samples [162]. The studies on single crystals, for example,
reveal a pronounced dependence of flux profile on the
two main parameters: the sample geometry [162, 167] and
the presence of structural or compositional defects [177]
(figure 9). While the sample geometry determines the
average domain structure characterized by regions with
uniform current densities and well-defined flux fronts the
defects usually define the regions of preferential flux
penetration, including no shielding or critical supercurrents.
The two patterns are competing such that, in cases of
large defect concentration, the flux and current profiles
are predominantly determined by defects (‘magnetically
induced granularity’) and the presence of extrinsic current
loops on various length scales [177]. Of special importance
are the studies of flux and current profiles in the
regions containing specific structural defects, such as grain
boundaries. The detailed results reported for [001] tilt
YBCO thin-film bicrystals [178] reveal characteristic and
reproducible cusp-like magnetic flux penetration in the
boundary region. The proposed model relates the cusp
structure to the intergranular critical current density which
sensitively depends, in turn, on misorientation angle of
a boundary. The information on its pinning properties
can be thus collected and correlated with its structural
features.
Current transfer and initial dissipation in HTSs
Figure 8. Schematic view of the Faraday effect and magneto-optical image of a rectangular, high-quality YBCO thin film.
The intermediate Al layer reflects the incident linearly polarized light. The magneto-optical layer rotates its polarization
vector, depending on the presence or absence of a local magnetic flux. Polarizer and analyser are set such that the
brightness of an imaged area is proportional to the intensity of the local magnetic flux. The zero-field-cooled film sample was
partially penetrated by the applied induction of 50 mT at 18 K. Discontinuity lines are clearly visible along diagonals.
Courtesy of Dr M Koblischka.
Figure 9. Magnetic flux distribution in BSCCO single
crystal at 2.5 K obtained immediately when the external
induction has been swept to 1 T. Dark areas correspond to
regions containing no penetrated flux. Although a pattern
with characteristic discontinuity lines can still be identified
the defects blur the flux distribution revealing
simultaneously an inhomogeneous distribution of bulk
shielding currents. Courtesy of Dr M Koblischka.
The studies of macroscopic (i.e. millimetre-scale)
current distributions represent perhaps the most impressive
demonstration of magneto-optical imaging potentials (figure 10).
A number of reports provide a quantitative reconstruction of inhomogeneous current paths in
thin [167] and thick [171] HTS films, as well as
BSCCO/Ag tapes [159, 179, 180] and multifilamentary
composites [181]. These studies demonstrate in particular
that the macroscopic critical current density Jc = Ic /A,
where A is the sample’s macroscopic (millimetre-scale)
cross-section, is determined by mesoscopic (micrometrescale) defects and related inhomogeneous local current distribution J (r). The spatially varying local current density
J (r) could be orders of magnitude higher than the average transport one (Ic /A), which directly proves the sizable
reduction of active current-carrying volume in real samples. This result is closely related to important fundamental
question, raised by a number of authors, on whether flux
pinning or percolation in a disordered network of dissipative centres (or centres of reduced current-carrying capabilities) determine the critical current in HTSs, in particular in the absence of or in small applied magnetic fields.
A recent elaboration of this dilemma, formulated in context of intended applications, may be found in [182]. In
BSCCO/Ag tapes direct evidence of current density variations was previously provided by slice-cutting experiments [183, 184] which demonstrate directly that there is
351
M Prester
Figure 10. (a ) Microstructure, (b ) image of the magnetic flux distribution and (c ), (d ) current stream lines for two intensities
of the applied magnetic field of BSCCO/Ag tape. In the darker regions near the silver–superconducting core interface in (c )
the gradient of penetrated flux is substantially steeper than in the central part of the tape, corresponding to higher local
values of the critical current density. Depending on tape quality, which varies along as well among the tapes, the local
current capacities vary [159] by a factor of 4. Closed macroscopic current loops in the central part, with sizes much above
the average grain size, introduce a very inhomogeneous current distribution and preferentially percolative current flow. The
effect is usually referred to [159] as magnetic granularity. Courtesy of Drs A Polyanskii and D Larbalestier.
352
Current transfer and initial dissipation in HTSs
a systematic increase of critical current density from the
centre toward the tape edge (figure 10). This result was attributed to well-aligned grains at the silver–superconducting
core interface. Also, it is worth mentioning that the concept of spatially non-uniform critical current distribution
(and its experimental counterpart [102, 103], d2 V /dI 2 ) is
supported by a large number of experimentally observed
transport features of bulk polycrystalline samples [103]. In
accordance with these findings the magneto-optical studies
of BSCCO/Ag tapes [159, 179, 180] provide not only visual
evidence but also a detailed quantitative elaboration of the
problem of non-uniform current densities. In these studies the magneto-optical layer was oriented perpendicular to
the plane of the tape-like sample and the magneto-optical
images revealed only the Hz (x, y) component of the penetrated magnetic flux. Under the assumption that the magnetization current J (r) induced by the applied field, co-planar
with the plane of the sample (and orthogonal to the highly
oriented c-axis), is effectively two dimensional and approximating the self-induced field H(r) by its major component,
Hz (x, y), the components of the two dimensional current
are given by
Jx (x, y) =
∂Hz
∂y
Jy (x, y) =
∂Hz
.
∂x
(8)
The current stream lines are therefore given by
contours of constant Hz (x, y), visualized in magnetooptical experiments, while the local current direction
coincides with tangents on contours [159], (figure 10).
The current paths were found to be very non-uniform
and sensitive to weak magnetic fields. Especially in
the central part of a sample, the arrays of macroscopic
current loops have been identified (magnetic granularity),
the global charge transport being obviously percolative
in nature.
It is important to note that the size of
these loops proves that they correspond to intergranular,
not intragranular, current. The regions of reduced and
highly inhomogeneous current coincide with areas of
increased defect concentration (e.g. processing-introduced
cracks [180]) and badly aligned grains. The magnetic
granularity can be substantially suppressed by application
of constant transport current [179], which indicates
that transport and magnetization determination of critical
currents involves different current patterns. The recent
studies on high-quality TBCCO thick films [171] employ an
advanced inversion scheme (involving fewer assumptions)
for reconstruction of the current paths.
In spite
of the almost perfect c-axis and considerable local
a–b texture the presence of similar inhomogeneities
in current transport has been demonstrated.
The
inhomogeneities may have different origins. Some of them,
revealing an abrupt reduction of local current, have been
attributed to intermittent colony boundaries characterized
by high misorientation. Apart from the latter source
of inhomogeneity, there is also an order-of-magnitude
variation in local (but still intergranular) currents which
may be attributed to general disorder in local intergranular
current capacity. The best local current density is up
to 10 times higher than the best transport result; the
latter itself varies by a factor of about 5 along the
film. Percolation, not flux pinning, has therefore been
identified [157, 159, 171, 182] as the main current-limiting
mechanism in available tapes and thick films of HTSs.
5. Discussion and conclusion
Considering the problem of current transfer in HTSs
in this review we have limited ourselves mainly to
various forms of polycrystalline HTS materials in small
applied magnetic fields. As documented by a number
of cited (and, perhaps, an even a larger number of
unfortunately not cited) experimental reports there is no
doubt that structural or compositional inhomogeneities,
primarily those related to grain boundaries, render the
current transfer inhomogeneous in these forms of HTS
samples. Numerous experiments have indeed demonstrated
not only the percolative character of current transfer but also
a consistency with universal laws of percolation and related
scaling phenomena. There are, however, an increasing
number of arguments suggesting that the relatively simple
case of weak-link networks of polycrystalline HTSs,
considered in this review, may also be applied as a
qualitative model for charge transfer in other transport
phenomena of HTSs. There are at least two relevant
examples worth discussing. The first concerns the problem
of vortex motion in various magnetic phases characterizing
the rich magnetic phase diagram [90] of microstructurally
(more) homogeneous HTSs (perfect single crystals, thin
epitaxial films). In spite the fact that drastic, orderparameter-breaking defects (such as grain boundaries) may
be absent in these samples, the omnipresence of disorder
may still lead to sort of ‘granularity’ inside the magnetic
vortex lattice: at fixed temperature some regions of
the vortex lattice can be well pinned while the other,
characterized by weak local pinning, can already be ‘free
to move’. The coexistence of these two types of regions
has been suggested [63, 185–190] to introduce the effective
percolation network inside the lattice of vortices resembling
the transport in its weak-link counterpart and classical
heterogeneous random networks. Numerous experimental
features in the mixed state of HTSs can be surprisingly
well interpreted, employing either the linear [187] or the
non-linear [188] limit of vortex response, by mapping
the problem of vortex motion on models of percolative
transport.
The other example for relevance of transport in
weak-link networks as a model system concerns the
problem of intrinsic transport of carriers in supposedly
ideal (boundary- and defect-free) single crystals and
epitaxial films. As is well known [5, 6], the metallic
phase occupies only a narrow compositional range of
the temperature–doping phase diagram; the overwhelming
area of the diagram corresponds to insulating phases.
The large areas of nominally insulating regions of the
diagram have been be attributed [191], in turn, to crossover regions inside which a property of the material
changes gradually with decreasing temperature.
The
two identified cross-over regions have been related to
the development of local antiferromagnetic correlations at
higher temperatures and to the formation of a pseudogap
353
M Prester
(i.e. just a suppression of the density of low-energy
excited states) at lower temperatures [191]. This suggests
that immediately below the pseudogap cross-over region
the [CuO2 ] planes initially develop local superconducting
correlations which ultimately (at temperature Tc ) evolve
into global superconducting order. In the regime of local
correlations one expects a pronounced role of percolation
between non-dissipative islands. Similar scenarios have
also been proposed by a phase separation model (see [143]),
a quantum percolation model [137, 138] and a model which
relies on intrinsic spatial inhomogeneities of HTSs [192].
Recent experimental investigations of transport in single
crystals and thin films [137, 193, 194] seem to favour the
percolation type of current transfer even in these close-toperfect forms of HTSs.
In summary, we reviewed the present understanding
of the problem of current transfer in HTSs, focusing on
the influence of intrinsic and extrinsic inhomogeneities
characterizing these materials. Considering primarily the
micrometre (mesoscopic) length scale of inhomogeneities
it was shown that a broad range of measurable transport
phenomena rely on non-uniform and percolative currents
as a crucial common ingredient characterizing the charge
transfer in real HTS samples.
References
[1] Tinkham M 1995 Introduction to Superconductivity
2nd edn (New York: McGraw-Hill)
Cyrot M and Pavuna D 1992 Introduction to
Superconductivity and High-Tc Materials (Singapore:
World Scientific)
[2] Ziman J M 1960 Electrons and Phonons (London: Oxford
University Press)
[3] Babcock S E and Vargas J L 1995 Annu. Rev. Mater. Sci.
25 193
[4] Ono A and Tanaka T 1987 Japan. J. Appl. Phys. 26 L825
[5] See, for example, Müller J and Olsen J L (eds) 1988 High
Temperature Superconductors and Materials and
Mechanisms of Superconductivity (Interlaken, 1988)
Physica C 153–155 (part I)
[6] See, for example, Wyder P (ed) Materials and
Mechanisms of Superconductivity, High Temperature
Superconductors IV (Grenoble, 1994) Physica C
235–240 (part I)
[7] Arlt J 1990 J. Mater. Sci. 25 2655
[8] Smith D A, Chisholm M F and Clabes J 1988 Appl. Phys.
Lett. 53 2344
[9] Chaudhary P and Matthews J W 1970 Appl. Phys. Lett. 17
115
[10] Sautter H, Gleiter H and Baro G 1977 Acta Metall. 24 467
[11] Zhu Y, Suenaga M and Sabatini R L 1994 Appl. Phys.
Lett. 65 1832
[12] Takahashi Y, Mori M and Ishida Y 1989 Appl. Phys. Lett.
55 486
[13] Tietz L A and Carter C B 1991 Physica C 182 241
[14] Ravi T S, Hwang D M, Ramesh R, Chan S W, Nazar L,
Chen C Y, Inam A and Venkatesan T 1990 Phys. Rev.
B 42 10 141
[15] Ekin J W, Braginski A I, Panson A J, Janocko M A,
Capone D W II, Zaluzec N J, Flandermeyer B,
de Lima O F, Hong M, Kwo J and Liou S H 1987 J.
Appl. Phys. 62 4821
[16] Gao Y, Merkle K L, Bai G, Chang H L M and Lam D J
1991 Physica C 174 11
[17] Camps R A, Evetts J E, Glowacki B A, Newcomb S B,
Somekh R E and Stobbs W M 1987 Nature 329 229
354
[18] Nakahara S, Fisanick G J, Yan M F, van Dover R B and
Boone T 1987 J. Crystal Growth 85 639
[19] Campbell A M 1989 Supercond. Sci. Technol. 2 287
[20] Parikh A S, Meyer B and Salama K 1994 Supercond. Sci.
Technol. 7 455
[21] Sathyamurthy S, Parikh A and Salama K 1996 Physica C
271 349
[22] Dimos D, Chaudhari P and Mannhart J 1990 Phys. Rev. B
41 4038
[23] Chaudhari P 1991 Physica C 185 292
Dimos D, Chaudhari P, Mannhart J and LeGoues F K
1988 Phys. Rev. Lett. 61 19
[24] Mannhart J 1990 J. Supercond. 3 281
[25] Chaudhari P, Mannhart J, Dimos D, Tsuei C C, Chi J,
Oprysko M M and Scheuermann M 1988 Phys. Rev.
Lett. 60 1653
[26] Mannhart J, Chaudhari P, Dimos D, Tsuei C C and
McGuire T R 1988 Phys. Rev. Lett. 61 2476
[27] Colclough M S, Gough C E, Keene M, Muirhead C M,
Thomas N, Abell J S and Sutton S 1987 Nature 328 47
[28] Ivanov Z G, Nilsson P A, Winkler D, Alarco J A,
Claeson T, Stepantsov E A and Tzalenchuk A Y 1991
Appl. Phys. Lett. 59 3030
[29] Mannhart J, Mayer B and Hilgenkamp H 1996 Z. Phys. B
101 175
[30] Hao L, Macfarlane J C and Pegrum C M 1996 Supercond.
Sci. Technol. 9 678 and references therein
[31] Weidel R, Brabetz S, Schmidl F, Klemm F, Wunderlich S
and Seidel P 1997 Supercond. Sci. Technol. 10 95
[32] Barone A and Paterno G 1982 Physics and Applications of
the Josephson Effect (New York: Wiley)
[33] Russek S E, Lathrop D K, Moeckly B H,Buhrman R
A, Shin D H and Silcox J 1990 Appl. Phys. Lett. 57
1155
[34] Likharev K K 1979 Rev. Mod. Phys. 51 101
[35] Delin K A and Kleinsasser A W 1996 Supercond. Sci.
Technol. 9 227
[36] Solymar L 1972 Superconducting Tunnelling and
Applications (London: Chapman and Hall)
[37] de Gennes P G 1964 Rev. Mod. Phys. 36 225
[38] Weertman J and Weertman J R 1966 Elementary
Dislocation Theory (New York: Macmillan)
Hirth J P and Lothe J 1982 Theory of Dislocations
(New York: Wiley)
[39] Chisholm M F and Pennycook S J 1991 Nature 351 47
[40] Cava R J 1990 Science 247 656
[41] Gross R and Mayer B 1991 Physica C 180 235
[42] Larbalestier D C, Babcock S E, Cai X Y, Field M B,
Gao Y, Heing N F, Kaiser D L, Merkle K,
Williams L K and Zhang N 1991 Physica C 185–189
315
[43] Strikovsky M, Linker G, Gaponov S, Mazo I and
Mayer O 1992 Phys. Rev. B 45 12 522
[44] Mayer B, Alff L, Trauble T and Gross R 1993 Appl. Phys.
Lett. 63 996
[45] Kawasaki M, Sarnelli E, Chaudhari P, Gupta A,
Kussmaul A, Lacey J and Lee W 1993 Appl. Phys. Lett
62 417
[46] Sarnelli E, Chaudhari P, Lee W Y and Esposito E 1994
Appl. Phys. Lett. 65 362
[47] Cardona A H, Suzuki H, Yamashita T, Young K H and
Bourne L C 1993 Appl. Phys. Lett. 62 411
[48] Nabatame T, Koike S, Hyun O B and Hirabayashi I 1994
Appl. Phys. Lett. 65 776
[49] Amrein T, Schultz L, Kabius B and Urban K 1995 Phys.
Rev. B 51 6792
[50] Wang J L, Tsu I F, Cai X Y, Kelley R J, Vaudin M D,
Babcock S E and Larbalestier D C 1996 J. Mater. Res.
11 868
[51] Babcock S E, Cai X Y, Kaiser D L and Larbalestier D C
1990 Nature 347 167
[52] Eom C B, Marshall A F, Suzuki Y, Boyer B,
Current transfer and initial dissipation in HTSs
Pease R F W and Geballe T H 1991 Nature 353 544
[53] Wang J L, Cai X Y, Kelley R J, Vaudin M D,
Babcock S E and Larbalestier D C 1994 Physica C 230
189
[54] Hampshire D P, Seuntjens J, Cooley L D and
Larbalestier D C 1988 Appl. Phys. Lett. 53 814
[55] Halbritter J 1992 Phys. Rev. B 46 14 861
[56] Halbritter J 1993 Phys. Rev. B 48 9735
[57] Ambegaokar V and Baratoff A 1963 Phys. Rev. Lett. 10
486
[58] Kulik I O and Omel’yanchuk A N 1975 Pis’ma Zh. Eksp.
Teor. Fiz. 21 216 (1975 JEPT Lett. 21 96)
[59] Babić E, Prester M, Marohnić Z, Car T, Biskup N and
Siddiqi S A 1989 Solid State Commun. 72 753
[60] Babić E, Prester M, Drobac D, Marohnić Z and Biskup N
1991 Phys. Rev. B 43 1162
[61] Moeckly B H, Lathrop D K and Buhrman R A 1993.
Phys. Rev. B 47(1) 400
[62] Robertazzi R P, Kleinsasser A W, Laibowitz R B,
Koch R H and Stawiasz K G 1992 Phys. Rev. B 46
8456
[63] Gurevich A 1990 Phys. Rev. B 42 4857
Gurevich A, Küpfer H and Keller C 1991 Europhys. Lett.
15 789
[64] Gurvitch M, Valles J M, Cucolo A M, Dynes J, Garno P,
Schneemayer L F and Waszczak 1989 Phys. Rev. Lett.
63 1008
[65] Gross R, Chaudhari P, Kawasaki M and Gupta A 1990
Phys. Rev. B 42 10 735
[66] Char K, Colclough M S, Lee L P and Zaharchuk G 1991
Appl. Phys. Lett. 59 2177
[67] Devaystov I A and Kupriyanov M Yu 1994 Pis’ma Zh.
Eksp. Teor. Fiz. 59 187 (1994 JETP Lett. 59 202)
[68] Tokura Y and Arima T 1990 Japan. J. Appl. Phys. 29
2388
[69] Quitmann C, Andrich D, Jarchow C, Fleuster M,
Beschoten B, Güntherodt G, Moschalkov V V,
Mante G and Manzke R 1992 Phys. Rev. B 46 11 813
and references therein
[70] Lee P A and Ramakrishnan T V 1985 Rev. Mod. Phys. 57
287
[71] Kaiser A B 1970 J. Phys. C: Solid State Phys. 3 410
[72] Kleiner R, Steinmayer F, Kunkel G and Müller P 1992
Phys. Rev. Lett. 68 2394
[73] Cho J H, Maley M P, Willis J O, Coulter J Y,
Bulaevskii L N, Haldar P and Motowidlo L R 1994
Appl. Phys. Lett. 64 3030
[74] Maley M P, Willis J O, Lessure H and McHenry M E
1990 Phys. Rev. B 42 2639
[75] Maley M P, Cho J H, Coulter J Y, Willis J O,
Bulaevskii L N, Motowidlo L R and Haldar P 1995
IEEE Trans. Appl. Supercond. 5 1290
[76] Stauffer D and Aharony A 1992 Introduction to
Percolation Theory (London: Taylor and Francis)
Bunde A and Havlin S 1991 Fractals and Disordered
Systems (Berlin: Springer)
Nakayama T, Yakubo K and Orbach R 1994 Rev. Mod.
Phys. 66 381
[77] Deutscher G 1988 Physica C 153–155 15
Clem J R 1988 Physica C 153–155 50
[78] See, for example, Gubser D U, Francavilla T L,
Leibowitz J R and Wolf S A (eds) 1980
Inhomogeneous Superconductors—1979 (AIP Conf.
Proc. 58) (New York: AIP)
Lobb C J and Frank D J 1984 Phys. Rev. B 30 4090
[79] Deutscher G, Imry Y annd Gunther L 1974 Phys. Rev. B
10 4598
[80] Lebeau C, Rosenblatt J, Raboutou A and Peyrel P 1986
Europhys. Lett 1 313
Rosenblatt J, Burin J P, Raboutou A, Peyral A and
Lebeau C 1991 Phase Transit. 30 157
[81] See, for example, England P, Goldie F and Caplin D A
1987 J. Phys. F: Met. Phys. 17 447
[82] England P, Venkatesan T, Wu X D and Inam A 1988
Phys. Rev. B 38 7125
[83] Tinkham M and Lobb C J 1989 Solid State Physics,
vol 42 (New York: Academic) p 91
[84] Marohnić Z and Babić E 1992 Magnetic Susceptibility of
Superconductors and other Spin Systems eds R A Hein,
T L Francavilla and D H Liebenberg (New York:
Plenum) p 267
[85] Prester M, Babić E, Stubičar M and Nozar P 1994 Phys.
Rev. B 43 2853
Prester M and Babić E 1994 Critical Currents in
Superconductors, Proc. 7th Workshop, ed H W Weber
(Singapore: World Scientific) p 521
[86] Bungre S S, Meisels R, Shen Z X and Caplin A D 1989
Nature 341 725
[87] Hascicek Y S and Testardi L R 1991 Phys. Rev. B 43
2853
[88] Prester M 1996 Phys. Rev. B 54 606
Prester M and Marohnić Z 1995 Phys. Rev. B 51 12 861
Prester M 1994 Physica C 235–240 3307
[89] Aponte J M, Bellorin A, Oentrich R, van der Kuur J,
Gutierrez G and Octavio M 1993 Phys. Rev. B 47 8964
Schneider T and Keller H 1993 Physica C 207 366
[90] Blatter G, Feigel’man M V, Geshkenbein V B, Larkin A I
and Vinokur V M 1994 Rev. Mod. Phys. 64 1125 and
references therein
[91] Huse D A, Fisher M P A and Fisher D S 1992 Nature
358 553
Fisher D S, Fisher M P A and Huse D A 1991 Phys. Rev.
B 43 130
[92] Jackson D J C and Das M P 1996 Supercond. Sci.
Technol. 9 713
[93] Leath P L and Tang W 1989 Phys. Rev. B 39 6485
[94] Rhyner Y and Blatter G 1989 Europhys. Lett. 10 401
Rhyner Y and Blatter G 1989 Phys. Rev. B 40 829
[95] Leath P L and Xia W 1991 Phys. Rev. B 44 9619
[96] Babić E, Prester M, Babić D, Nozar P, Stastny P and
Matacotta F C 1991 Solid State Commun. 80 855
[97] Manhhart J, Gross R, Hipler K, Huebner R P, Tsuei C C,
Dimos D and Chaudhari P 1989 Science 245 839
[98] Soulen R J Jr, Francavilla T L, Fuller-Mora W W,
Miller M M, Joshi C H, Carter W L, Rodenbush A J,
Manlief M D and Aized D 1993 Phys. Rev. B 50 478
[99] Ambegaokar V and Halperin B I 1969 Phys. Rev. Lett. 22
1364
[100] Tinkham M 1988 Phys. Rev. Lett. 61 1658
[101] Dubson M A, Herbert S T, Calabrese J J, Harris D C,
Patton B R and Garland J C 1988 Phys. Rev. Lett. 60
1061
[102] Warnes W H and Larbalestier D C 1986 Cryogenics 26
643
[103] Babić E, Prester M, Biskup N, Marohnić Z and
Siddiqi S A 1990 Phys. Rev. B 41 6278
Babić E, Prester M, Drobac D, Marohnić Z, Nozar P,
Stastny P, Matacotta F C and Bernik S 1992 Phys. Rev.
B 45 913
[104] Sun J Z, Eom C B, Lairson B, Bravman J C and
Geballe T H 1991 Phys. Rev. B 43 3002
[105] Kosterlitz J M and Thouless D J 1973 J. Phys. C 6 1181
[106] Lobb C J, Abraham D W and Tinkham M 1983 Phys.
Rev. B 27 150
[107] Kim D H, Goldman A M, Kang J H and Kampwirth R T
1989 Phys. Rev. B 40 8834 and references therein
[108] Halperin B I and Nelson D R 1979 J. Low Temp. Phys. 36
599
[109] Mikheenko P N, Dou S X, Lewandowski S J and
Abaliosheva S I 1996 Supercond. Sci. Technol. 9 942
and references therein
Miu L, Jakob G, Haibach P, Kluge Th, Frey U,
Voss-de Haan P and Adrian H 1998 Phys. Rev. B 57
3144
355
M Prester
[110] Repaci J M, Kwon C, Li Q, Jiang X, Venkatesan T,
Glover R E III and Lobb C J 1996 Phys. Rev. B 54
R9674
[111] Langer J S and Fisher M E 1967 Phys. Rev. Lett. 19 560
[112] Senoussi S 1992 J. Phys. III 2 104 and references therein
[113] Cesnak L, Gömöry F and Kovač 1996 Supercond. Sci.
Technol. 9 184
[114] Zeldov E, Amer N M, Koren G, Gupta A, Gambino R J
and Elfresh M W 1989 Phys. Rev. Lett. 62 3093
[115] Tkazczyk J E, Arendt R H, Garbauskas M F, Hart H R,
Lay K W and Luborsky F E 1992 Phys. Rev. B 45
12 506
[116] Kovač P, Cesnak L, Melišek T, Hušek I and Frölich K
1997 Supercond. Sci. Technol. 10 605
[117] Koch R H, Foglietti V, Gallagher W J, Koren G, Gupta A
and Fisher M P A 1989 Phys. Rev. Lett. 63 1511
Gammel P L, Schneemeyer L F and Bishop D J 1991
Phys. Rev. Lett. 66 953
Charalambous M, Koch R H, Masselink T, Doany T,
Feild C and Holtzberg F 1995 Phys. Rev. Lett. 75 2578
[118] Worthington T K, Olsson E, Nichols C S, Shaw T M and
Clark D R 1991 Phys. Rev. B 43 10 538
Tiernan T T M, Joshi R and Hallock R B 1993 Phys. Rev.
B 48 3423
Ziq Kh A, Abdelhadi M M, Hamdan N M and
Al Harthi A S 1996 Supercond. Sci. Technol. 9 192
[119] Hamzic A, Fruchter L and Campbell I A 1990 Nature 345
515
[120] Minnhagen P and Olsson P 1992 Phys. Rev. B 45 5722
[121] Sugano R, Toshiyuki O and Murayama Y 1993 Phys. Rev.
B 48 13 784
[122] Sergeenkov S A 1990 Physica C 167 339
[123] Horng H E, Jao J C, Chen H C, Yang H C, Sung H H and
Chen F C 1989 Phys. Rev. B 39 9628
[124] Meisels R, Bungre S and Caplin A D 1988
J. Less-Common Met. 151 83
[125] Peyral P, Lebeau C, Rosenblatt J, Raboutou A, Perrin C,
Pena O and Sergent M 1989 J. Less-Common Met. 151
49
[126] Bungre S S, Cassidy S M, Caplin D A, Alford N McN
and Button T W 1991 Supercond. Sci. Technol. 4 S250
[127] Isaac I and Jung J 1997 Phys. Rev. B 55 8564 and
references therein
[128] Granato E and Domínquez 1997 Phys. Rev. B 56 14 671
[129] Leemann C, Flückiger Ph, Marisco V, Gavilano J L,
Srivastava P K, Lerch Ch and Martinoli P 1990 Phys.
Rev. Lett. 64 3082
[130] Ebner C and Stroud D 1983 Phys. Rev. B 28 5053
[131] Goldschmidt D 1989 Phys. Rev. B 39 9139
[132] Gerber A, Grenet T, Cyrot M and Beille J 1990 Phys.
Rev. Lett. 65 3201
Gerber A, Grenet T, Cyrot M and Beille J 1991 Phys.
Rev. B 43 12 935
[133] Ausloos M, Clippe P and Laurent Ch 1990 Phys. Rev. B
41 9506
[134] Pureur P, Costa R M, Rodrigues P Jr, Schaf F and
Kunzler J V 1993 Phys. Rev. B 47 11 420
[135] Khalil A E 1996 Philos. Mag. B 73 619
[136] Mott N F and Davis A D 1979 Electronic Processes in
Non-Crystalline Materials 2nd edn (Oxford:
Clarendon)
[137] Okunev V D, Pafomov N N, Svitsunov V M,
Lewandowski S J, Gierlowski P and Kula W 1996
Physica C 262 75
[138] Dallacasa V and Feduzi R 1995 Physica C 251 156 and
references therein
[139] Shklovskii B I and Efros A L 1984 Electronic Properties
of Doped Semiconductors (Springer Series in Solid
State Sciences) ed M Cardona, P Fulde and
H-J Quiesser (Berlin: Springer)
[140] Sheng P and Klafter J 1983 Phys. Rev. B 27 2583
[141] Roy D, Ghosh B, Neogy C, Deb S K and Nag A 1995
356
Phys. Status Solidi b 190 511
[142] Kallio A and Xiong X 1991 Phys. Rev. B 43 5564
[143] Sigmund E and Müller K A (eds) 1994 Phase Separation
in Cuprate Superconductors (Berlin: Springer)
[144] Efros A L and Shklovskii B I 1975 J. Phys. C 8 L49
[145] Cattaneo L, Celasco M, Masoero A, Mazzeti P, Puica I
and Stepanescu A 1996 Physica C 267 127
[146] Bobyl A V, Gaevski M E, Khrebtov I A, Konnikov S G,
Shantsev D V, Solovev V A, Suris R A and
Tkachenko A D 1995 Physica C 247 7
[147] Kiss L B and Svedlindh I 1994 IEEE Trans. Electron.
Devices 41 2112
[148] Ferrari M J, Johnson M, Wellstood F C, Clarke J, Mitzi D,
Rosenthal P A, Eom C B, Geballe T H, Kapitulnik A
and Beasley M R 1990 Phys. Rev. Lett. 64 72
[149] Kiss L B and Svedlindh P 1993 Phys. Rev. Lett. 71 2817
[150] Prester M and Marohnić Z 1993 Phys. Rev. B 47 2801
[151] Bulaevskii L N, Clem J R, Glazman L I and
Malozemoff A P 1992 Phys. Rev. B 45 2545
[152] Bulaevskii L N, Daemen L L, Maley M P and
Coulter J Y 1993 Phys. Rev. B 48 13 798
[153] Hensel B, Grivel J-C, Jeremie A, Perin A, Pollini A and
Flükiger R 1993 Physica C 205 329
Hensel B, Grasso G and Flükiger R 1995 Phys. Rev. B 51
15 456
[154] Umezawa A, Feng Y, Edelman H S, Willis T C,
Parrel J A, Larbalestier D C, Riley G N and
Carter W L 1994 Physica C 219 378
[155] Li Y H, Kilner J A, Dhalle M, Caplin A D, Grasso G and
Flükiger R 1995 Supercond. Sci. Technol. 8 764
[156] Eibl O 1995 Supercond. Sci. Technol. 8 883
[157] Goyal A, Specht E D, Kroeger D M, Mason T A,
Dingley D J, Riley G N Jr and Rupich M W 1995
Appl. Phys. Lett. 66 2903
[158] Caplin A D, Cohen L F, Cuthbert M N, Dhalle, Lacey D,
Perkins G K and Thomas J V 1995 IEEE Trans. Appl.
Supercond. 5 1864
[159] Pashitski A E, Polyanskii A, Gurevich A, Parrell J A and
Larbalestier D C 1995 Physica C 246 133
[160] Miu L and Popa S 1996 Physica C 265 43
[161] Dhalle M, Cuthbert M, Johnston M D, Everett J,
Flükiger R, Dou S X, Goldacker W, Beales T and
Caplin A D 1997 Supercond. Sci. Technol. 10 21
[162] Koblischka M R and Wijngaarden R J 1995 Supercond.
Sci. Technol. 8 199 and references therein
[163] Oral A, Bending S J, Humphreys R G and Henini M 1997
Supercond. Sci. Technol. 10 17 and references therein
[164] Hess H F, Robinson R B, Dynes R C, Valles J M and
Waszczak J V 1989 Phys. Rev. Lett. 62 214
Hess H F, Murray C A and Waszczak J V 1992 Phys.
Rev. Lett. 69 2138
[165] Andreev K E, Bobyl A V, Gudushnikov S A,
Karmanenko S F, Krasnosvobodtsev S L,
Matveets L V, Snigirev O V, Suris R A and
Vengrus I I 1997 Supercond. Sci. Technol. 10 366
[166] Brandt E H and Indenbom M 1993 Phys. Rev. B 48
12 893 and references therein
[167] Schuster T, Kuhn H and Indenbom M V 1995 Phys. Rev.
B 52 15 621
Schuster T, Kuhn H, Brandt E H, Indenbom M V,
Koblischka M R and Konczykowski M 1994 Phys.
Rev. B 50 16 684 and references therein
[168] Roth B J, Sepulveda N G and Wikswo J P Jr 1989
J. Appl. Phys. 65 361
[169] Brandt E H 1992 Phys. Rev. B 46 8628
[170] Xing W, Heinrich B, Zhou H, Fife A A and Cragg A R
1994 J. Appl. Phys. 76 4244
[171] Pashitski A E, Gurevich A, Polyanskii A E,
Larbalestier D C, Goyal A, Specht E D, Kroeger D M,
DeLuca J A and Tkaczyk J E 1997 Science 275 367
[172] Wijngaarden R, Spoelder H J W, Surdeanu R and
Griessen R 1996 Phys. Rev. B 54 6742
Current transfer and initial dissipation in HTSs
[173] Tanihata A, Sakai A, Matsui M, Nonaka N and
Osamura K 1996 Supercond. Sci. Technol. 9 1055
[174] Wijngaarden R J, Koblischka M R and Griessen R 1994
Physica C 235-240 2699
[175] Dorosinskii L A, Indenbom M V, Nikitenko V I,
Ossip’yan Yu A, Polyanskii A A and
Vlasko-Vlasov V K 1992 Physica C 203 149
[176] Indenbom V M, Forkl A, Ludescher B, Kronmüller H,
Habermeier H U, Leibold B, D’Anna G, Li T W,
Kes P H and Menovsky A A 1994 Physica C
226 325
[177] Koblischka M R 1996 Supercond. Sci. Technol. 9 271
[178] Polyanskii A A, Gurevich A, Pashitski A E, Heinig N F,
Redwing R D, Nordman J E and Larblestier D C 1996
Phys. Rev. B 53 8687
[179] Pashitski A E, Polyanskii A, Gurevich A, Parrell J A and
Larbalestier D C 1995 Appl. Phys. Lett. 67 2720
[180] Parrell J A, Polyanskii A, Pashitski A E and
Larbalestier D C 1995 Supercond. Sci. Technol.
9 393
[181] Welp U, Gunter D O, Crabtree G W, Zhong W,
Balachandran U, Haldar P, Sokolowski R S,
Vlasko-Vlasov V K and Nikitenko V I 1995 Nature
376 44
[182] Lubkin G 1996 Phys. Today 49 48
[183] Larbalestier D C, Cai X Y, Feng Y, Edelman H,
Umezawa A, Riley G N Jr and Carter W L 1994
Physica C 221 299
[184] Grasso G, Hensel B, Jeremie A and Flükiger R 1995
Physica C 241 45
[185] Wördenweber R 1992 Phys. Rev. B 46 3076
[186] Prester M 1995 Quantum Dynamics in Submicron
Structures ed H A Cerdeira, B Kramer and G Schön
(Dordrecht: Kluwer) p 645
[187] Ziese M 1996 Phys. Rev. B 53 12 422
Ziese M 1996 Physica C 269 35
[188] Ziese M 1997 Phys. Rev. B 55 8106
[189] Yamafuji K and Kiss T 1996 Physica C 258 197
[190] Jagla E A and Balseiro C A 1996 Phys. Rev. B 53
R538
[191] Batlogg B and Emery V J 1996 Nature 382 20
[192] Phillips J C 1996 Proc. Soc. Photo-Opt. Instrum. Eng.
2697 2
Phillips J C 1995 Phys. Rev. B 51 15 402 and references
therein
[193] Grigelionis G, Kundrotas P J, Tornau E E and
Rosengren A 1996 Supercond. Sci. Technol. 9
927
[194] Lemor W, Wübbeler G and Schirmer O F 1995 Solid
State Commun. 95 823
357