Rep. Prog. Phys. 60 (1997) 1581–1672. Printed in the UK
PII: S0034-4885(97)41466-5
Open questions in the magnetic behaviour of
high-temperature superconductors
L F Cohen† and Henrik Jeldtoft Jensen‡
† Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK
‡ Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, UK
Received 10 March 1997
Abstract
A principally experimental review of vortex behaviour in high-temperature superconductors
is presented. The reader is first introduced to the basic concepts needed to understand the
magnetic properties of type II superconductors. The concepts of vortex melting, the vortex
glass, vortex creep, etc are also discussed briefly. The bulk part of the review relates the
theoretical predictions proposed for the vortex system in high temperature superconductors
to experimental findings. The review ends with an attempt to direct the reader to those
areas which still require further clarification.
c 1997 IOP Publishing Ltd
0034-4885/97/121581+92$59.50 1581
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Contents
1. Introduction
2. The vortex system and its behaviour
2.1. Type I and type II superconductors
2.2. Isotropic ideal type II superconductors
2.3. H c1 and H c2
2.4. Disturbances of the ideal hexagonal flux-line lattice
2.5. Fluctuations in the order parameter
2.6. Anisotropy
2.7. Thermal equilibrium and non-equilibrium properties
2.8. Symmetry of the order parameter
3. Interpreting vortex behaviour
3.1. Transport and magnetization measurements
3.2. Transport E–J curves
3.3. Flux creep
3.4. Magnetic measurement analysis
3.5. Critical scaling applied to E–J curves
4. Experimental observation of vortex behaviour
4.1. Reversible properties
4.2. The irreversibility line
4.3. In the vicinity of H irr
4.4. The order of the melting transition
4.5. Below the irreversibility line—the vortex solid
5. Summary of the questions at the brink of resolution
Acknowledgments
References
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Magnetic behaviour of superconductors
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1. Introduction
The justification for a predominantly experimental review of the magnetic behaviour in hightemperature superconductor (HTS) materials is simply the recognition that an introductory
text, as this sets out to be, may yet help shed light on the behaviour of vortices in the presence
of disorder. Furthermore, if we are to utilize HTS materials in the form of magnets, power
cables or high-frequency filters, summarizing our understanding of the pinning properties of
vortices and the form of the HT diagram in equilibrium or otherwise—is still of paramount
importance.
Several excellent reviews have recently appeared (Farrell 1994, 1995, Fischer 1993,
1994, Blatter et al 1994b, Brandt 1995) and inevitably there will be some overlap.
Interpretation of the experimental evidence will probably not stand the test of time in
the same way as a theoretical review because as new data appears it sheds light on all that
has gone before. Nevertheless, we feel it is important to brave the unknown, take a snap
shot in time, examine the current position critically and ask whether various types of novel
behaviour which have been predicted have indeed been observed.
Our task is made more difficult because observed vortex behaviour is linked to
underlying static disorder and the general classification of material purity and quality is
still incomplete. In an attempt to simplify matters we have restricted the discussion to
single crystals and, where necessary, to thin films. We limit the discussion further to
YBCO 123 and BSCCO 2212 to illustrate the range of properties in systems with very
different anisotropy.
In reality the magnetic properties of HTS is a labyrinth. One can easily get lost
and confused. The sirens’ song sounds seductively in the form of wonderful sparkling
theoretical inventions: Bose glass, vortex glass, quantum creep, melting, entanglement,
disentanglement, pancakes, dimensional crossovers, plastic flow. The list goes on and on.
Only a strong and clearly directed guide will enable one to make it through the maze. Even
more so if one not only wants to survive the expedition, but in fact has the ambition to
refresh one’s mind and gain overview and understanding from the quest.
In section 2 we develop the basic notions used to describe magnetic properties of
superconductors. We then introduce various concepts used to describe HTS behaviour
which are novel at least to superconductivity, and highlight those ideas which are simply an
extension of concepts discussed previously. We discuss similarities and differences between
conventional and HTS. The main difference between the new and old superconductors is the
relevance of thermal fluctuations and strong anisotropy.We have tried to address in general
terms the connection between the new theoretical developments and quantities measured
in experiments. We pay special attention to the importance of distinguishing between
thermodynamic equilibrium quantities and non-equilibrium experiments.
In section 3 we review some concepts related to the interpretation of experimental data
which we feel would otherwise weigh down the discussion in the next section. We place
significant emphasis on the understanding of transport and magnetization measurements as
these provide the bulk of experimental evidence.
In section 4 we really start our expedition into the wilderness. For consistency sake we
have tried to divide the enormous body of experimental information into broad headings
which reflect the novel phase diagrams which have been proposed for HTS materials. We
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L F Cohen and H J Jensen
review evidence for the behaviour of Hc1, Hc2, the form and meaning of the irreversibility
line, the melting line and glassy behaviour. We discuss the different regions of behaviour
in the field (H ) and temperature (T ) plane emerging from transport and magnetization data,
in order to explore whether different experimental angles of approach lead to a consistent
picture. The phenomenology derived from this exploration is brought into contact with
various the theoretical pictures introduced in section 2.
In section 5 we summarize the open questions which remain.
2. The vortex system and its behaviour
In this section we will run through the basic concepts of how a magnetic field behaves
inside a superconductor. Details are elaborated in many books on superconductivity. Two
excellent classics are Tinkham (1995) and de Gennes (1966). We will also expand on these
ideas in very general terms to include the novel aspects of HTS in magnetic fields. A further
reference which gives a good idea of the complexity of the problem is the extensive review
by Blatter et al (1994b).
2.1. Type I and type II superconductors
Superconductors exist in one of two types. In the first kind an external magnetic field cannot
penetrate into the bulk of the sample without destroying the superconducting condensate.
We are not going to deal more with this kind. The second kind of superconductors, of
which HTS are prominent members, are able to remain superconducting over a range of
fields H in the interval Hc1 < H < Hc2 . At the lower critical field Hc1 the first magnetic
flux starts to enter the bulk of the superconductor. The field does not penetrate the bulk in a
homogenous way. Had this been the case, the magnetic properties of type II superconductors
would have been much simpler. The mixed state which exists for fields between Hc1 and
Hc2 is spatially inhomogeneous. Both the local magnetic induction and the local density of
superconducting electrons are position dependent. The magnetic field penetrates the bulk
of the superconductor in the form of quantized flux tubes or magnetic vortices.
2.2. Isotropic ideal type II superconductors
Figure 1 illustrates a vortex line and the important lengths λ, the penetration depth and ξ ,
the coherence length. At zero temperature in an isotropic ideal superconductor containing
no inhomogeneities in the superconducting matrix the mixed state is threaded by straight
vortex lines running parallel to the direction of the external magnetic field. They are called
vortex lines because they consist of vortices in the superfluid of Cooper-paired electrons.
These vortices were discovered by Abrikosov using the phenomenological Ginzburg–Landau
theory (Abrikosov 1957). The diverging circulation velocity as one approaches the centre
of the vortex drives the density of superelectrons to zero. At the axis of the vortex the
superconducting order parameter is equal to zero. It increases as one goes radially out from
the vortex core and reaches its asymptotic limit over a distance given by the Ginzburg–
Landau coherence length ξ . For HTS, ξ ≈ 10–20 Å at zero temperature (in the direction
parallel to the superconducting planes, see below). At the centre of the vortex the magnetic
induction is maximum. This field is screened by√the circulating supercurrents. As a result
the magnetic induction decreases as exp(−r/λ)/ r as one goes away from the vortex axis.
The field and circulating currents decrease rapidly to zero beyond the London penetration
depth λ. Each vortex carries one quantum of magnetic flux φ0 = h/2e where h is Planck’s
Magnetic behaviour of superconductors
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Figure 1. An illustration of a vortex line and the important lengths, λ the penetration depth and
ξ the coherence length.
constant and e is the elementary charge. Therefore the number of flux lines inside the
sample is approximately proportional to the external field. For HTS, λ ≈ 1500 Å at zero
temperature. The circulating supercurrents (or equivalently the magnetic induction) give
rise to an interaction between the flux lines, extending out to a distance of order λ. The
depletion of the order parameter at the vortex axis also leads to a short-range attraction
between vortices. The long-range magnetic interaction is repulsive for straight parallel
vortex lines and attractive for antiparallel lines. Due to this interaction the minimum energy
configuration for a flux-line system in an ideal isotropic superconductor consists of parallel
vortex lines arranged in an hexagonal lattice in the plane perpendicular to the field direction.
2.3. Hc1 and Hc2
The transitions at Hc1 and Hc2 can be thought of as follows. At the lower critical field the
(Gibb’s) free energy of the state without a flux line is equal to the state with one flux line
(or many non-interacting flux lines). It uses energy to keep the field out of the bulk of
the superconductor. As soon as the field becomes a tiny bit larger than Hc1 flux lines will
flow into the bulk. As long as they do not interact, the free energy is independent of the
number of flux lines within the bulk. Hence, the flux lines will rush in unhindered until
they start to have an average separation of order λ. Accordingly the magnetic field at Hc1
will approximately be one flux quantum within the area of a circle of radius λ. The precise
expression of Hc1 is Hc1 = φ0 ln(λ/ξ )/(4πλ2 ). As the external field is gradually increased,
the flux lines are squashed together. Eventually their cores, in which the superconducting
order parameter is equal to zero, will begin to overlap and the whole bulk becomes normal.
This is what happens at Hc2 , at least at the simplest mean-field Ginzburg–Landau level of
description. We expect the value of Hc2 to be given by one flux quantum through an area of
the size of the core. This simple picture becomes more complicated as the effect of thermal
fluctuations is included. Fluctuations always become important close to the temperature
where the system undergoes a (continuous) phase transition. The width of the fluctuation
regime depends on the ‘stiffness’ of the order parameter close to Tc (H ). Mean-field theory
is applicable as long as the length scale of spatial variations is longer than or equal to the
coherence length ξ of the mean-field theory. When the thermal energy kB T becomes of
the order of the free energy within a correlated volume ξ 3 f , where f is the free-energy
density, fluctuations make the mean-field theory inappropriate. Since ξ ∼ (Tc − T )−1/2 and
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L F Cohen and H J Jensen
f ∼ (Tc − T )2 close to Tc , in mean-field theory ξ 3 f/kB T ≈ 1 as Tc is approached (Landau
and Lifshitz 1969). Precisely how close to Tc fluctuations become important depends on
the size of ξ and Tc .
For conventional low-temperature superconductors ξ is large and Tc is small. The
transition in these superconductors is therefore described well by mean-field Ginzburg–
Landau theory except for an unresolvably narrow region of width about 10−4 K around
Tc . In the HTS the situation is reversed. The coherence length is short and the transition
temperature is high. Thermal fluctuations are accordingly much more important over a
broad regime around the Hc2 (T ) line. In conventional superconductors the experimentally
observed transition from a resistive phase for T > Tc (H ) to a phase with zero resistivity
below Tc (H ) occurs very sharply as the temperature is lowered through the meanfield transition temperature Tc (H ). In HTS the resistivity only vanishes slowly while a
superconducting state gradually builds up. The true superconducting phase transition takes
place at a temperature significantly lower than the mean-field value for Tc (H ). Below we
shall return to the question: What replaces the mean-field Hc2 (T ) line?
The very nature of the Hc2 (T ) line is unclear when one goes beyond mean-field
considerations. It is not even known if fluctuations change the nature of the transition from a
continuous transition (as in mean field) to a first-order transition. Some calculations suggest
that the transition in the pure system is first order and that disorder replaces the transition
by a gradual crossover from the vortex liquid to a vortex solid (Moore and Newman 1995).
Also the nature of the mixed state in the neighbourhood of the Hc2 (T ) is much more subtle
than hitherto anticipated. This is also true for conventional superconductors. However,
for the low-temperature superconductors these questions are mainly of non-observational
academic interest. In HTS this subtle neighbourhood is much broader and of greater practical
importance.
2.4. Disturbances of the ideal hexagonal flux-line lattice
Let us again return to the simple mean-field Ginzburg–Landau description of
superconductivity. We need to consider the extent to which the ideal hexagonal line lattice
can be disturbed by defects and thermal fluctuations.
2.4.1. Defects. Inhomogeneities in the superconducting material can lead to a local
suppression of the superconducting order parameter. An example is a void or a hole in
the superconductor. This will lead to an interaction between the vortex core and void. In
order to minimize the suppression of the order parameter it will be beneficial to locate
the vortex core on top of the void, thereby only depleting the order parameter once. This
mechanism leads to core pinning. The void attracts the vortex line, hence it tends to trap
or pin the line. The spatial variation in the flow pattern of the supercurrents can also lead
to pinning, especially to surface pinning. Core pinning is in general the most important
bulk pinning mechanism. A random arrangement of material inhomogeneities will lead to
a random potential and random forces acting on the vortex lines. This will disturb the
positional order of the flux lattice.
The interaction between the inhomogeneities in the superconducting matrix and the flux
lines is of crucial importance for the ability of the superconductor to support a dissipation
free electric current when penetrated by vortices. The reason is as follows. As an electric
current is passed through the superconductor the Lorentz force will act between electrons
and the (localized) magnetic field carried by the flux lines. In this absence of pinning
this force will move the flux lines with the result that a time-dependent local magnetic
Magnetic behaviour of superconductors
1587
Figure 2. An illustration of the defect structure and the soft and stiff vortex lattice ‘trying’ to
take advantage of the defects.
induction is established. As a result, an electric field is induced which then acts on the
electrons thereby leading to a voltage and corresponding dissipation. The details of this
scenario are controlled by the Josephson relation (Josephson 1965, Tinkham 1995). The
only way dissipation can be avoided when a magnetic field threads the superconductor is by
preventing the flux lines from moving. Since inhomogeneities attract the flux lines they are
able to prevent this motion up to a certain pinning force Fp . The degree to which the flux
lines are pinned determines the maximum Lorentz force one can apply without dissipation.
The Lorentz force (per volume) is given by Bj , where B is the magnetic induction locally
averaged over the flux lines and j is the current density. The maximum dissipation free
current—called the critical current—is given by jc = Fp /B. It is important to keep in mind
that even if one could make the pinning centres infinitely strong there would still be an
upper bound for the dissipation free current. Dissipation will then occur when the Cooper
pairs start to break up due to the induced current. This happens at the depairing current
where the kinetic energy of the Cooper pairs equals to the superconducting condensation
energy which binds the electrons together in Cooper pairs (see Tinkham 1995). However,
in a magnetic field the depinning critical current is always found to be smaller than the
depairing current.
The efficiency of the pinning centres depends indirectly on the strength of the vortexvortex interaction. A very stiff vortex system will not be able to adjust to the random pinning
potential and can therefore not relax deeply into the pinning potential. A soft vortex system
on the other hand will be able to adjust itself to the random pinning forces and thereby sink
deep down into the pinning potential. This leads to a more strongly pinned configuration
than in the case of a stiff vortex system. Figure 2 illustrates point defect structure and soft
and stiff vortex lattices attempting to fit to it.
The interaction between flux lines can, as a good approximation, be described by a
two-body potential between flux-line elements. For an isotropic interaction, the interaction
energy between two flux-line segments dl1 and dl2 separated by the distance r12 is given
by the sum of two exponentially screened Coulomb-like contributions
0
0
Um + Uc = dl1 · dl2 e−r12 /λ − |dl1 ||dl2 |e−r12 /ξ .
(2.1)
The first term has its origin in the magnetic field carried by the flux lines. This interaction
1588
L F Cohen and H J Jensen
√
is screened beyond an effective field-dependent magnetic penetration depth λ0 = λ/ 1 − b.
Where b = B/Bc2 is the ratio between the actual induction B and the induction Bc2
corresponding at the upper cirtical field Hc2 . The second term describes
an attraction between
√
the core of the flux lines and is very short ranged ξ 0 = ξ/ 2(1 − b). The interaction
between flux-line elements in anisotropic superconductors is of the same nature although
in more complicated detail due to the dependence of the interaction on the orientation of
r12 with respect to the symmetry axis of the material. An excellent discussion of these
important details is given by Brandt (1995) in his recent review.
The interaction between the flux lines is responsible for the elastic rigidity of the flux
lattice. The elastic properties are described by a shear C66 modulus, a tilt modulus C44 , and a
compression modulus C11 . These moduli have been calculated for isotropic superconductors
as well as for anisotropic superconductors, see again Brandt (1995). Here we list the
expressions of elastic moduli. An essential point to bear in mind is that these moduli are
field dependent such that C11 ∼ C44 ∼ b2 and C66 ∼ b(1 − b)2 , where b = B/Bc2 . The
shear modulus vanishes at the upper critical field giving rise to a softening of the flux lattice
and thereby a more effective pinning close to Hc2 . The increase in the pinning force close to
Hc2 is known as the peak effect (Pippard 1969). Another important point is that the tilt and
compression moduli depends strongly on the wavelength of the imposed elastic deformation.
A short wavelength tilt deformation uses significantly less energy than homogeneous tilt.
This has to be taken into account when making quantitative estimates of the deformations
of the ideal flux lattice.
The interaction between a single pinning centre and a segment of a vortex line is difficult
to calculate and depends on the nature of the pinning interaction. However, it is useful to
bear in mind an estimate of the pinning energy obtained from the excluded volume effect.
If the defect depresses the superconducting order parameter in a volume of size V (smaller
than the core volume) the energy gained by positioning the core of the vortex line on top
of the defect will be of order VHc2 ,√since the superconducting condensation energy density
is given by the square of Hc = φ/2 2πλξ . Close to Hc2 the condensation energy vanishes
as (1 − b) (Thuneberg 1984). Since C66 vanishes as (1 − b)2 the pinning energy may
dominate over the elastic energy in this field regime. This is the explanation of the peak
effect mentioned above and discussed in sections 4.3.2 and 4.5.1.
Pinning of the flux lines is not only induced by random disorder in the bulk. Any spatial
inhomogeneity in the superconducting properties may make the free energy of a vortex line
depend on position. The energy of the supercurrents circulating the vortex line will vary
close to the sample surface. This effect leads to the Bean–Livingston surface barrier which
a flux line parallel to a plane surface has to surmount in order to enter into the bulk of the
sample (see e.g. de Gennes 1966). Another barrier to flux entry relates to the shape of the
sample and is denoted by a geometric barrier (see e.g. Zeldov et al 1994). The barrier is
again an energy barrier that the flux line will have to overcome in order to move between
the interior and exterior of the sample. The barrier is estimated from two contributions.
One is the energy needed to create a flux line of the length of the sample thickness. The
other contibution is the energy extracted when the Lorentz force induced by the circulating
supercurrents, perform work on the flux line while these currents attempt to move the flux
line towards the centre of the sample. If one chose an appropriate shape of the sample
this barrier can by made to vanish. See figure 16. For further discussion see sections 4.2.2
and 4.5.1.
Finally we must mention that the layered structure of the cuperate HTS leads to a
significant intrinsic pinning. When the flux lines are arranged parallel to the superconducting
copper-oxide planes energy is gained when the normal core of the flux line is positioned
Magnetic behaviour of superconductors
1589
in the less strongly superconducting region in between the superconducting planes. The
flux lines have to overcome an essential energy barrier in order to move across the
superconducting planes. This effect leads to dramatic peaks in the critical currents when
field alignment is nearly parallel to the copper-oxide planes. See figure 17.
2.4.2. Thermal fluctuations. Melting of the flux lattice. As discussed in section 2.3, thermal
fluctuations are much more important in HTS than in low-temperature superconductors.
Thermal fluctuations can also perturb the flux-line configuration. Like in an ordinary
crystal lattice kept at a constant temperature, elastic forces in the vortex lattice will result in
displacements, u, of the vortex lines away from the ideal configuration to reach an average
distance given by hu2 i ∝ T . The proportionality constant is determined by the elastic moduli
of the flux lattice. If there is sufficient thermal energy available so that u becomes of order
10–30% of the average flux-line separation, one can expect that the flux-line lattice might
melt (Nelson 1988). This is called the Lindemann melting criterion U = cL a0 , here cL is
somewhere around 0.1–0.3 and a0 denotes the average flux-line separation. The criterion is
a phenomenological principle that is known to work for ordinary crystals. The limitation
of this principle is that it does not explain what type of fluctuations in the lattice structure
(dislocations, disclinations etc) actually causes the melting. One should of course find a
way to calculate the temperature at which the shear modulus describing homogeneous shear
goes to zero (the short wavelength shear modulus remains non-zero in the liqiud). This has
not yet been done (for any three-dimensional system in fact). Nevertheless, the melting of
the flux-line lattice is thought of as a melting in the traditional sense, namely as a softening
of the flux lattice leading to a phase unable to support any shear.
The curve Tm (H ) in the T –H phase diagram at which the flux lattice melts is called
the melting line. In principle for fields very close to Hc1 the flux system should always be
a ‘liquid’ since the flux lines are separated more than λ leading to a vanishing interaction
between the vortices. In practice, flux lines enter the sample at Hc1 very rapidly and
the region where the separation is larger than the interaction length is hardly accessible.
However, very recently this ‘re-entrant’ melting effect has been observed experimentally
(Ravikumar 1997) in the low-temperature superconductor Nb2 Se.
The fact that the melted flux-line lattice has lost its rigidity (shear modulus equal to zero)
has been used more or less intuitively to suggest that dissipation, i.e. flux flow, is more easy
in the melted phase than in the solid phase. It does not need to be so. One should keep in
mind the fact that the solid Abrikosov lattice will flow subject to the slightest applied driving
force if no pinning potential is present to hinder the motion. Furthermore, the efficiency of
the pinning potential is reduced by the stiffness of the flux system. In order to follow the
pinning centres, the flux lattice has to distort. This uses elastic energy. Non-interacting flux
lines of infinite flexibility would be able to take full advantage of the pinning potential. A
decrease in the elastic moduli is a move towards this optimum situation. The peak in the
pinning force observed close to Hc2 in conventional superconductors was once connected
with the softening of the elastic coefficients of the flux-line lattice with field (Pippard 1969,
Larkin and Ovchinninkov 1979).
If the shear modulus of the flux lattice vanishes upon thermal melting the pinning centres
could act more efficiently leading to a reduced mobility of the flux lines and therefore a
reduced dissipation. This scenario assumes a density of pinning centres much higher than
the flux-line density. If there are only a few very strong pinning centres the situation would
be reversed. The melted flux liquid will be able to flow in between the pinning centres
leading to an increased dissipation above the melting temperature. In any case, one assumes
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L F Cohen and H J Jensen
that the energy scale of the pinning potential is larger than the available thermal energy kB Tm
at the melting transition. Otherwise the pinning centres would already have become unable
to trap the flux system at a lower temperature. In this case one would observe a depinning
line (see section 2.4.3) rather than a melting line.
2.4.3. Irreversibility line. Real systems always contain pinning centres. This leads to yet
another energy scale. The properties of the flux system are determined by the relationship
between the thermal energy Eth , the vortex–vortex energy Evv , and the pinning energy Epin .
The competition between these three energy scales is complicated because the effective
pinning energy depends on the vortex–vortex interaction energy. Pinning centres become
more effective if the interaction between vortices is small.
The competition between Eth and Epin leads to the existence of the irreversibility line
in the H –T plane. This line is determined as follows. For temperatures below this line
the pinning is strong enough to be able to trap flux lines as they are pushed in and out of
the sample. If one sweeps the external field from zero up to a value H > Hc1 and down
again to zero some flux lines will remain pinned inside the sample even after the external
field has returned to zero. The magnetization of the sample behaves in an irreversible
way. Above the line the only contribution to the magnetization comes from the reversible
Meissner effect. The thermal energy now dominates the pinning energies so that the flux
is no longer trapped in metastable configurations. The irreversibility line is identified in
current–voltage experiments as the line in the H –T plane that separates the region of fields
and temperatures above the line where the voltage depends linearly on the applied current
from a region below the line of a nonlinear current–voltage characteristics. Figure 3 shows
the rough effective pinning potential acting on a flux bundle. Eth = kB T is the scale of the
thermal fluctuation. Flux bundle at position A is easily thermally activated out of the local
potential well. The flux bundle at B is trapped.
In theory, there is no reason for a specific relationship to exist between the irreversibility
line and the melting line. The melting occurs when Eth ≈ Evv . Reversible behaviour is
separated from irreversible behaviour when Eth ≈ Epin . Depending on the accidental relation
between Epin and Evv (accidental since Epin depends on the properties of the defects of the
Figure 3. The rough effective pinning potential acting on a flux bundle. Eth = kB T is the scale
of the thermal fluctuation. The flux bundle at position A is easily thermally activated out of the
local potential well. The flux bundle at B is trapped.
Magnetic behaviour of superconductors
1591
material, whereas Evv is an intrinsic flux lattice property) the melting line and irreversibility
line can be very close together, or melting can occur at a lower or higher temperature
than where reversibility sets in. All three possibilities have been identified in experiments
although the different lines were originally assumed to be the same.
2.4.4. Vortex glass. The effect of the pinning potential (which results from the underlying
static disorder) is at the heart of the approach to the phases of the flux system that has
become known as the vortex glass scenario (Fisher 1989). The vortex glass approach tries
to incorporate the pinning potential from the beginning. This is in contrast to the melting
theory described above (see section 2.4.2). The melting theory focuses on the properties of
the pure system. The effect of the pinning potential is then treated as a perturbation of the
pure system.
The high-temperature phase of the vortex glass model is considered to be a liquid of
mobile flux lines moving unhindered over the pinning potential. The low-temperature phase
is an immobile amorphous phase. Because the pinning potential is supposed to disorder the
flux system. The essence of the vortex glass picture is that collective effects are predicted to
be able to produce infinite energy barriers leading to a strictly zero linear flux-flow resistance
as J approaches zero. Different workers have emphasized different aspects of the physics
of the vortex glass. Fisher (1989) introduced the term vortex glass. Fisher was especially
concerned with the transition between the high-and low-temperature phase. Fisher argued
that the transition is a true phase transition. Furthermore, Fisher assumed the transition
to be continuous and worked out a scaling theory for the voltage–current characteristics in
the vicinity of the transition (see section 3.5 below). Feigel’man et al (1989) formulated a
theory of the voltage–current characteristics inside the low-temperature ‘glass’ phase. They
generalized the collective pinning approach developed by Larkin and Ovchinnikov (1979) to
discuss the collective behaviour of the flux system. Their model is known as the ‘collective
creep’ theory. Fiegel’man et al (1989) calculated from elasticity theory the effective energy
barriers set up by the competition between the elastic vortex–vortex interaction and the
pinning potential. They considered how the flux bundles creep over these barriers and they
derived power laws for the logarithmic time dependence of the electric current inside the
vortex glass regime. Experimental observation of the vortex system deep in the vortex solid
is discussed in section 4.5.
Various viewpoints are advocated concerning the thermal stability of the vortex glass.
One school (Nelson and Vinokur 1992, 1993) claimed that a thermodynamically stable glass
is only possible if strong disorder is present in the form of columnar defects or twin planes.
Randomly positioned point defects might induce distortions of the flux lattice but point
defects will not be strong enough to establish a stable glass in the thermodynamic sense.
The vortex glass idea was originally suggested in connection with point defects. It was
suggested (Fisher et al 1991) that point defects are able to produce the diverging barriers
associated with the vortex glass. Experimentally this question is delicate since the response
of a system with large but finite barriers might easily look like the response from a system
with infinite barriers. The situation is similar to the one encountered in connection with
ordinary glasses. As one goes through the glass temperature the viscosity changes by 15
orders of magnitude. However, so far no one has been able to show that this change is
related to a genuine phase transition (see e.g. Nagel 1993).
The vortex glass is supposed to be a consequence of diverging energy barriers (Fisher
et al 1991, Feigel’man et al 1989). That such a divergence may in principle exist is most
simply seen from the following argument. The relaxation of the flux-line structure is always
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L F Cohen and H J Jensen
caused by a driving (Lorentz) force which is proportional to the local current density j . This
current arises due to the existence of a gradient in the density of the flux lines. According
to the Maxwell relation ∇ × B = j , i.e. j goes to zero as the flux structure relaxes to a
homogeneous arrangement. Assume that the Lorentz force is able to move a flux bundle of
volume V . The total driving force on this volume, Fd , is proportional to Vj . The driving
force has to be able overcome the pinning force acting on this volume. The pinning force,
Fp , is a sum of V np (here np is the density of pinning centres) individual pinning forces
acting in random directions. We estimate the sum of the pinning forcespby measuring the
variance of the sum of V np independent random numbers, i.e. Fp ∼ V np (Larkin and
Ovchinnikov 1979). Precisely when the driving force is able to make the flux bundle inside
the volume V move we have Fd = Fp and therefore V ∼ 1/j 2 . Hence, as j → 0 the volume
that will have to move coherently increases. This makes the energy barrier produced by the
pinning centres and the compressibility of the flux system (Feigel’man et al 1989) diverge.
It is always difficult to establish equilibrium in systems with large energy barriers. This
is a well known theme in the field of spin glasses. This must be kept in mind when analysing
experiments. The properties observed below the irreversibility line, where the vortex glass
is supposed to exist, are hysteretic (by definition) and their relation to genuine equilibrium
properties is complicated.
Another point to keep in mind is that the diverging energy barrier arises because it is
assumed that the coherent motion of a larger and larger volume is necessary in order to
induce relaxation. This may not be so. Plastic deformations of the flux structure may be
able to break the flux bundle volume up into smaller pieces which can be moved by a
finite-energy input. The scenario is easy to visualize in two dimensions. Here the energy
of a dislocation moving through the flux lattice is some finite energy given by the shear
strength of the system and expected to be proportional to the shear constant C66 . (For
a three-dimensional flux system this energy will be proportional to the thickness of the
sample.) When the barrier needed to move the increasing coherent volume V becomes
larger than the dislocation barrier the volume will break up into sub-volumes separated by
boundaries of sliding dislocations. The scenario is more subtle in three dimensions where
it is more difficult to visualize the nature of the plastic deformation that may cut up the
coherent volume. One possibility is that the flux lines cut through each other. In this case
the diverging energy barrier will be replaced by the energy scale of flux cutting.
At this point it is important to note that vortex glass scaling behaviour can only be
observed in a restricted regime. Namely, in the current regime where the volume V (j ) is
increasing with decreasing current density j . However, when the energy barrier associated
with this volume becomes larger than the plastic barrier the increase in the energy barrier
with decreasing j is cut off for currents below some current scale jplas . Lack of resolution
in experiments may make it difficult to probe the current scales below jplas . One can then
be misled into the false conclusion that the system exhibits a vortex glass transition. A
similar difficulty is encountered in numerical simulations of weak pinning centres where it
may be difficult to reach system sizes larger than the volume V (jplas ) associated with the
onset of plastic deformations.
2.4.5. Plastic flow. Plastic deformations are also of crucial importance at the depinning
transition. The effect is most clearly seen at zero temperature. Consider a pinned flux
system under application of a transport current, or driving force Fd = Bj . When the
driving force equals the volume pinning force, the flux structures start to break away from
the pinning centres. The onset of motion can either take place as a coherent displacement
Magnetic behaviour of superconductors
1593
of the entire flux strucure or in the form of incoherent motion of parts of the flux array
in between islands of pinned vortices. In the latter case plastic shearing of the flux lattice
obviously occur. This scenario has long been observed in experiments (Wördenweber et
al 1986, Wördenweber and Kes 1986, Bhattacharya and Higgins 1993, and Yaron et al
1995), and was for instance considered theoretically by Kramer (1973). In principle plastic
shearing will always occur if the flux system is large enough. This is clearly seen in finitesize scaling studies of computer simulations (Jensen et al 1988a) where rather picturesque
channel-flow patterns were observed.
The size dependence of the onset of plastic flow is most easily understood by a clear
mean-field argument due to Coppersmith and Millis (1991). Consider the balance between
the forces acting on a volume V = Ld . Here L is the linear dimension of the d-dimensional
volume. There are three different types of forces acting on the volume. Namely, the applied
force Fa induced by the applied current. Secondly, the pinning force Fp produced by the
pinning centres within the volume V . And finally, the vortex–vortex interaction force Fb
acting across the boundary of the volume between the vortex structure inside the volume
V and the rest of the vortex structure outside this volume. When the applied force is tuned
precisely to the threshold for depinning (i.e. the applied current equals the critical current)
these three forces are exactly at balance with each other Fa = Fb + Fp . The boundary force
Fb = Fd − Fp is needed to compensate the mismatch between the globally averaged pinning
force, which is the threshold force Fthr and the local pinning force Fp , which fluctuates
from one sub-volume to another. The deviation between the actual sum of the random
pinning forces inside the volume V and the global average will scale as the square root of
the number of pinning centres contained in the volume V , see section 2.4.4. Accordingly
we have Fb ∼ Ld/2 . There are Ld−1 bonds across the boundary of the volume. Therefore
the force fb that each individual bond has to support will scale as fb ∼ L1−d/2 , i.e. the
load on the individual bonds increases with the size of the volume which is assumed to
act coherently. Since a bond will only be able to support a stress up to a certain value,
the coherent volume will break up into smaller pieces.
√ Thus, the threshold for the onset
of plastic deformations is expected to scale like 1/ L in one dimension. A logarithmic
size dependence is expected in two dimensions. This size scaling agrees qualitatively with
simulations in one and two dimensions (Jensen et al 1988a, Jensen 1995). In three and
higher dimensions the plastic onset of deformations will also occur. However, one has to go
beyond the simple average arguments presented here and consider rare events (Coppersmith
and Millis 1991).
2.5. Fluctuations in the order parameter
In section 2.3 we mentioned that HTS are much more susceptible to thermal fluctuations
than low-temperature superconductors. The specific nature of the fluctuations in the order
parameter is not yet completely clear.
One type of fluctuation is similar to the fluctuations known to occur in two dimensions.
For very thin films (which can be modelled as two-dimensional systems) of conventional
superconductors it has been know for many years that the superconducting transition is
completely controlled by thermal fluctuations (Minnhagen 1987). Since the sample is very
thin, the thermal energy close to Tc is able to create vortex excitations. So instead of inducing
vortices by an external magnetic field vortex pairs can be thermally excited. Somewhat like
the appearance of bubbles in water just below the boiling temperature.
In three-dimensional samples vortex-loop excitations play the role of the twodimensional vortex–antivortex excitations. In fact the temperature dependence of the
1594
L F Cohen and H J Jensen
resistivity near Tc led Minnhagen (Persico et al 1996) to conclude that the transition (in
zero-magnetic field) in bulk HTS may be controlled by unbinding vortex loops by cutting
through single superconducting planes.
The role of vortex-loop excitations in a non-zero magnetic field have been intensively
studied by computer simulations (Chen and Tietel 1995, Caneiro 1995, Nguyen et al 1996)
as well as analytically (Tes̆anović 1995).
Even without identifying the nature of the fluctuations one can derive a relation for
the temperature dependence of the magnetic field at the phase transition. In zero field the
symmetry of the Ginzburg–Landau free energy is the same as the symmetry of the threedimensional XY model. The critical exponents of the XY model are well known. The
correlation length, for instance, diverges like ξ ∼ 1/|T − Tc |ν , where ν ≈ 0.66, when Tc is
approached. We can now attempt to deduce the shift in the transition temperature produced
by an applied magnetic field B. The field B, the flux quantum φ0 and the correlation length
ξ , can be combined in a dimensionless combination like Bξ 2 /φ0 . From this we conclude
that the field at which the transition occurs, must scale like B ∼ ξ −2 ∼ |T − Tc |2ν . Where
Tc is the transition temperature in the zero-magnetic field. It is not clear how large magnetic
fields can be applied before this scaling relation for B(T ) may break down. However, we
shall see below (section 4.2.2) that a relationship B(T ) ∼ |T − Tc |4/3 is in fact consistent
with several experiments.
2.6. Anisotropy
Most conventional superconductors are isotropic or only weakly anisotropic. The cuprate
superconductors are layered structures (perovskites) and therefore inherently anisotropic.
The degree of anisotropy varies enormously for the different types of HTS. We will
concentrate on YBCO 123 as an example of the superconductors with the smallest anisotropy
and we choose BSSCO 2212 as an example of the strongly anisotropic samples. The
anisotropy gives rise to new effects. First it makes the flux system more susceptible to
finite wavelengths tilt. More dramatically the anisotropy, which has its origin in the layered
nature of the HTS materials, may lead to a dimensional crossover. This crossover can be
viewed as a change from a situation where the flux system can be treated as consisting of
continuous flux lines to a situation where the layered structure of the materials manifests
itself more explicitly. Quantitatively, the crossover takes place when the superconducting
coherence length perpendicular to the layers, ξ⊥ , becomes of the order of the distance
between the layers (Klemm et al 1975).
The energy of the continuous flux lines depends on their orientation relative to the
crystal axis. This situation is described by a Ginzburg–Landau free energy in which the
gradient term is anisotropic. As the effective anisotropy becomes stronger this description
is replaced by a free-energy functional describing a set of superconducting layers coupled
together via Josephson coupling (Lawrence and Doniach 1971). The continuous flux lines
are replaced by stacks of two-dimensional vortices confined to the superconducting layers
but coupled across the layers by to the Josephson effect (for a review see Fischer 1993,
1994).
2.7. Thermal equilibrium and non-equilibrium properties
The properties of the magnetic flux system inside the superconductor must be divided into
two categories: equilibrium and non-equilibrium properties. Furthermore, it is important to
distinguish between static and dynamic properties.
Magnetic behaviour of superconductors
1595
Among the equilibrium properties one would first like to establish the phase diagram
of the flux system as a function of field and temperature. The melting line in the ideal
system without any pinning potential, has attracted much attention (Nelson 1988). The
structural character of the flux system, i.e. the order transverse to the field direction, and the
order along the field direction, in the different phases should be determined. The dynamic
properties of systems without pinning are simple. As soon as a current is passed through the
material, the Lorentz force will make the flux system flow with a velocity proportional to
the current. This results in a current-independent flux-flow resistance (Bardeen and Stephen
1965, Tinkham 1995). In the absence of pinning the melted flux system flows in the same
way as a flux lattice when a constant uniform Lorenz force is applied. Only if one applies
a Lorentz force that varies in space will the difference between the finite shear rigidity of
the flux solid and the zero shear modulus of the liquid flux system show up in transport
experiments.
The presence of a pinning potential may dramatically change the situation. The pinning
potential disturbs the translational order of the flux-line lattice. Even weak pinning can
make it difficult to experimentally access the thermal equilibrium states. One signature
of this is the observed history dependence of the established flux structure. Recent highprecision neutron scattering experiment on the flux-line lattice in a clean niobium sample
by Gammel et al (1994) found that the best orientationally ordered flux-line lattice was
established by entering the superconducting state by slowly decreasing the magnetic field
through Hc2 rather than field cooling or zero-field cooling followed by an increase of the
magnetic field (Mason 1991).
The irreversibility line itself separates non-equilibrium configurations for temperatures
below the irreversibility line from those above. The Bean critical state (see Tinkham 1995)
is a non-equilibrium state and relaxation of the magnetization in this state occurs as the
flux lines creep towards equilibrium, as illustrated in figure 4. The rate of relaxation has
been viewed as a signature of the specific phase the relaxing flux system is in (Malozemoff
and Fisher 1990, Krusin-Elbaum et al 1992). Since one is dealing with an intrinsically
non-equilibrium property one must exercise great care in this approach.
A similar situation is encountered when one wants to deduce the nature of the phase
of the flux system or the nature of the transition between flux phases from transport
measurements. The resistance obtained from voltage–current measurements has been used
to conclude that the melting of the flux-line system is a first-order transition in clean samples
and that the transition becomes second order as pinning becomes relevant (Safar et al 1992c).
It has only recently (Jiang et al 1995) been appreciated that the hysteretic behaviour of the
Figure 4. An illustration of the Bean flux profiles across a sample during a sweeping up of the
external magnetic field in (a) the absence of thermal activation and (b) the presence of thermal
activation. The broken curve indicates the profile after some time has elapsed.
1596
L F Cohen and H J Jensen
resitivity might be due to non-equilibrium effects rather than the assumed first-order nature
of flux-line lattice melting.
The out-of-equilibrium driven flux-line lattice has properties of its own that cannot
be directly related to the phase of the non-driven equilibrium system. One example is
the tendency to ordering of the flux lattice subject to a driving force. As mentioned in
section 2.4.5, the spatial fluctuations in the random pinning force can tear the flux lattice
into pieces when the applied current is close to the depinning current. As the current is
increased the flux lattice is ‘lifted out’ of the pinning potential. This reduces the effect of
the pinning forces. The forces acting between the flux lines may then be able to induce
more order into the lattice structure than would be observed in the pinned non-driven system
in equilibrium. The ordering due to an applied current was observed long ago in neutron
scattering experiments (Thorel 1973). Recently, Koshelev and Vinokur (1994) suggested
that the applied current might induce a phase transition from a disordered moving system,
for currents close to the depinning current, to a moving flux system with crystalline order
at larger currents. Although it is clear that some ordering occurs as the drive is increased
much current work attempts to clarify the precise details of the nature of the ordering of
the moving system (Giamarchi and Le Doussal 1996, Moon et al 1996, Faleski et al 1996,
Spencer and Jensen 1997).
2.8. Symmetry of the order parameter
The symmetry of the Ginzburg–Landau wavefunction, which describes the motion of the
superconducting charges, may be different to that of conventional superconductors. The
standard symmetry is the spherically symmetric s-wave, with the notation of atomic orbital
theory. Experimental and theoretical studies suggest that in HTS this spherical symmetry
is replaced by a ‘four-leaf clover’ symmetry characteristic of d-orbital. Since the symmetry
of the superconducting wavefunction is reflected in the symmetry of the vortex core the
microscopic symmetry of the Ginzburg–Landau wavefunction may influence the properties
of the vortex system (see, e.g. Berlinsky et al 1995). In these circumstances one finds that the
ideal hexatic Abricosov lattice is modified. The structure of the vortex lattice depends on the
magnetic field and the degree of assumed mixing between the s-wave and d-wave component
of the Ginzburg–Landau wavefunction. The larger the magnetic field (or the stronger the
d-wave component) the more the vortex lattice changes structure from a triangular lattice
towards a square lattice. Note that this will influence the elastic moduli of the flux lattice.
Another important difference between the s-wave superconductor and d-wave
superconductor is the difference in the quasiparticle spectrum around the vortex line. The
density of states around the vortex in the d-wave has four-fold symmetry (Schophol and
Maki 1995). The energy gap for creating quasiparticle excitations in the d-wave vanishes
along certain directions of the quasiparticle momentum. As a consequence, the temperature
dependence of, say, the magnetic penetration depth and specific heat will be different in the
d-wave superconductor compared with the s-wave case.
3. Interpreting vortex behaviour
3.1. Transport and magnetization measurements
The two main experimental approaches to studying vortex behaviour are direct transport
measurements and magnetic studies. The electric field E within the superconductor is
explicitly measured in the former approach, but plays a less obvious though no less important
Magnetic behaviour of superconductors
1597
role in the latter. This makes a comparison of the data from each technique difficult.
Vibrating reed, AC susceptibility, mechanical oscillator and torque magnetometry all belong
in the second category but will not be discussed specifically.
In a four-terminal transport measurement it is usual to apply current and to measure
a voltage. Electric field (E) versus current density (J ) curves can then be plotted as a
function of temperature T or magnetic field B. The local slope ρ = dE/dJ of the E–J
curve at a fixed voltage (threshold voltage) can be extracted and plotted as a function of
the current density. The limitations of transport measurements are related to the practical
constraints of attaching contacts, passing large currents through those contacts and measuring
small voltages (10−8 V). Consequently transport measurements are usually made at high
temperatures or large magnetic fields, where the loss (voltage) is significant.
The magnetic measurement has the advantage of being contactless. The magnetic
moment can be translated into a current density, but care is needed to take into account
sample geometry effects. The electric field is set by the experimental conditions, most
usually the sweep rate of the external magnetic field, or more generally by the rate of
change of flux through the sample. The local slope of the E–J curve can be obtained from
a magnetization measurement in a number of ways. For example, small changes in the
magnetic field sweep rate (or electric field) alter the induced current through the sample and
variation of current as a function of sweep rate is a measure of the local slope of the E–J
curve. (This is also later referred to as the dynamic creep rate.)
In theory, magnetic and transport techniques yield identical information. However, at
the same fixed magnetic field the transport measurements are restricted to high temperatures
and the magnetization to intermediate and low temperatures. As described by Caplin et al
(1994), an E–J –B surface can be constructed which schematically illustrates the regimes
which each kind of measurement is capable of accessing. It is extremely rare to find
transport and magnetization measurements made in overlapping regions of the E–J –T –B
parameter space.
The effective well depth for pinning centres Ueff (J ), depends on the current density
J such that when J = Jc , Ueff (Jc ) = 0. In a transport measurement, currents in the
vicinity of the critical value can be applied and the onset of irreversible behaviour (nonohmic E–J curve), can be examined in detail. The transport measurement is used to study
the reversible state close to Tc and it is also frequently used to examine the nature of the
reversible–irreversible (or solid-to-liquid) transition. In a magnetization measurement the
electric fields are lower than in the transport measurement. The irreversible magnetization
(or current density) reflects the Bean profile which is set up across the sample. This means
that the current density is always less than the critical value and for a given temperature
and electric field the critical value can only be approached by increasing the magnetic field.
In order to examine the behaviour and transformations deep inside the pinned vortex solid
the magnetization measurement is the obvious choice. Figure 5 illustrates the variation of
the effective pinning well with applied current.
Transport and magnetic measurements can yield complementary information, but this
is not always the case. The reason is associated with the relative strengths of the vortex
pinning forces and driving force. When the pinning forces are weak compared with the
driving forces, the vortex lattice deforms elastically and flows coherently, as often observed
in a transport measurement. When the two forces are comparable plasticity and incoherent
motion results. This is more frequently observed in magnetic measurements. Once the
pinning potential dominates, thermally activated creep is anticipated and again the system
may behave elastically. Only in this case may one expect direct complementarity between
transport and magnetic measurements.
1598
L F Cohen and H J Jensen
Figure 5. A variation of the effective pinning well Up (r) with applied current I . The well is
tilted by the current and for I = Ic , the local minima in Up (r) vanishes.
3.2. Transport E–J curves
In this section we discuss the essential mechanisms responsible for the different types
of current–voltage (I –V ) curves one can observe in superconductors. According to the
Josephson relation (Tinkham 1995) motion of the vortices inside a superconductor gives
rise to an electric field between two points A and B proportional to the number of vortices
crossing the line connecting A and B per unit of time.
Let us first consider the zero-temperature case. The disorder in the superconducting
matrix can prevent the motion of the vortices as long as the applied Lorentz force produced
by the applied current is smaller than a depinning current jc corresponding to the pinning
force. The E–J characteristics are therefore of the following form:
(
0
if J < Jc
E=
f (J )
if J > Jc .
Since eventually all vortices must depin and flow with a velocity proportional to J we
must always have f (J ) ∝ J for J Jc . The shape of the function f (J ) for currents
in the vicinity above Jc depends on the dimension of the system (Larkin and Ovchinnikov
1986) and on the topology of the induced flux motion (Jensen et al 1988b). The flux
system can either depin as a coherent structure moving homogeneously through the pinning
potential. The pinning force will deform the elastic flux structure but only elastically.
Alternatively, motion can occur inhomogeneously when the flux system depins. In this case
plastic deformations will take place when the more mobile parts of the flux array passes
by the stronger pinned, and therefore less mobile, regions of flux lines. The latter scenario
appears to be the most common (Wördenweber et al 1986, Bhattacharya and Higgins 1993,
Yaron et al 1995).
In two dimensions f (J ) ∝ J if all vortices depin at the same value of J . A nonlinearity
in f (J ) arises due to partial depinning as J is increased leading to successively more
vortices being pulled into the flow (Jensen et al 1988b) our two-dimensional simulation. The
situation is less simple in three dimensions. The nonlinearity can here contain contributions
both from elastic distortions of the flux lattice as it flows and from inhomogeneous plastic
flow of the flux lines (Larkin and Ovchinnikov 1986, Bhattacharya and Higgins 1993).
The above picture is changed at non-zero temperature due to thermal activation of the
Magnetic behaviour of superconductors
1599
vortices over the pinning barriers. This can lead to a non-zero electric field even for J < Jc .
The actual shape of the I –V curve for currents J 6 Jc is conveniently discussed in terms
of the following ansatz
E = Bv
(3.1)
where v is the flux velocity with the following exponential activation form
v = ωd exp[−Ueff (J, B, T )/kB T ].
(3.2)
The prefactor ω denotes the ‘attempt frequency’, i.e. the number of times the effective flux
bundle tries to overcome the barrier per unit time. The factor d denotes the distance moved
by the flux bundle as it jumps over the barrier. The product ωd can be measured, but it is
difficult to determine each factor separately.
Equation (3.1) follows from the Josephson relation which can be expressed in the form
E = φ0 nv where φ0 is the flux quantum and n is the density of the vortices which move
with velocity v.
The detailed dependence of Ueff on J will determine the shape of the I –V curve. The
behaviour of Ueff in the limit of J approaching zero is of special importance. If the barrier
remains finite Ueff (J ) → U0 as J → 0 equation (3.1) leads to a linear I –V at small current.
To obtain this result one must remember to include the contribution in equation (3.1) of flux
bundles jumping in the direction opposite to the applied force as well as those jumping with
it. Since the applied current lowers the barrier for jumps in the direction of the force and
increases the barrier for jumps in the opposite direction we can write the effective barrier
for small currents as Ueff = U0 ± J ∂Ueff (0)∂J . Subtracting the contribution of the jumps
up against the applied force from the contribution produced by the jumps in the direction
of the applied force leads to the famous result (Tinkham 1995, Blatter et al 1994b)
E ≈ v exp[−U0 /T ]2J ∂Ueff (0)∂J.
(3.3)
We conclude that the I –V curve has three regions of different behaviour depending on
the sizes of the current compared with the zero-temperature critical current Jc (see figure 6).
When J < Jc the electric field is either linear in J (finite barriers) or E vanishes faster than
linear when J → 0 (infinite barriers). In the region of currents J ≈ Jc a rapid increase
in E is bound to take place as the barriers vanish. Finally, for J > Jc one enters the free
flux-flow regime where the current is able to completely overcome the pinning potential.
In this regime E ∝ J .
3.3. Flux creep
The specific purpose of studying the vortex solid deep inside the irreversible regime, is to
determine the functional dependence of Ueff (B, J, T ), the effective pinning well depth, and
compare it with theoretical predictions to understand which barriers are involved in different
regions of the H –T plane and for different kinds of static disorder.
An electric field is associated with a time-varying supercurrent and all thermally
activated forward vortex motion can be described by the simple rate equation.
−Ueff (J )
Bd
−Ueff (J )
dJ
exp
∝E=
= Bωd exp
(3.4)
dt
t0
kT
kT
where t0 is the hopping attempt time for a vortex or vortex bundle, ω is the effective attempt
frequency and d is the hop distance. The rate equation can be solved approximately once
the form of Ueff (J ) is known.
1600
L F Cohen and H J Jensen
Figure 6. A schematic illustration of three regions of the I –V curve shape at (a) T = 0 and
(b) finite T .
There are many experimental artefacts such as irregular sample shapes, temperature
instability and magnetic field stability (transients in the magnet itself), which can influence
the form of the decay. Recently, spatial relaxation information using Hall probe arrays, have
confirmed that global measurements can only be simply interpreted once the DC magnetic
fields is greater than the penetration field and in general at fields where surface or geometric
barriers are insignificant (Brawner et al 1993b, Abulafia et al 1995).
Anderson–Kim thermally activated flux creep. This simple model assumes thermal
activation of uncorrelated vortices or vortex bundles over a net potential barrier which
depends linearly on applied current density J :
J
(3.5)
Ueff = Uc 1 −
Jc
where Uc is the J = 0 barrier and Jc is the current density in the absence of thermal
activation.
Combining equations (3.4) and (3.5)
t
kT
ln
.
(3.6)
J (T , t) = Jc 1 −
Uc
t0
To eliminate the unknown Jc in equation (3.5), it is convenient to evaluate a normalized
relaxation rate defined as the logarithmic derivative of the magnetization or current density
Magnetic behaviour of superconductors
1601
(in the Bean critical state the magnetization is proportional to the current density J )
S=−
−d ln(J )
−d ln(M)
1 d(J )
≈
=
.
J d ln(t)
d ln(t)
d ln t
(3.7)
Long-time relaxation. The most common way to examine flux creep by magnetic
measurement is to study the long-time decay of a magnetization signal. The magnetic
field is applied to the sample at a certain rate (dH /dt), setting a certain electric field
through the sample. The field sweep is stopped at a desired value of magnetic field. As
discussed by Gurevich et al (1991, 1993), an initial ‘settle time’ has to be allowed for while
the electric field spatially redistributes through the sample as a consequence of the change
of driving conditions from the initial voltage driven to the final current driven situation.
The decay of magnetization dM/dt, now sets the electric field across the sample and can
in principle be translated as a journey down the E–J curve. If the decay is approximately
logarithmic in time, the slope of dlnM/dlnt is approximately equivalent to the inverse slope
of the ln E–ln J curves over the range of electric fields associated with the experiment. For
further details see Zhukov (1992).
From equation (3.6), we can write
−1
kT
kT
ln(t/t0 )
.
(3.8)
1−
S=
Uc
Uc
Abulafia et al (1995, 1997) developed an analysis based on the rate equation which
allows a direct model-independent determination of the local activation energy and
logarithmic timescale t0 for flux creep.
Dynamic relaxation. Dynamic relaxation is a technique developed by Pust (1990) and
Pozek et al (1991). It monitors the change in the magnetization signal as a function of the
sweep rate of the magnetic field. The normalized dynamic sweep rate Q is defined as
Q=
d ln M
.
d ln(dH /dt)
(3.9)
The equivalence of Q and S has been discussed by Jirsa et al (1993) and Schnack et al
(1992). The general case for I –V curves of different shapes has been analysed by Zhukov
(1992).
Deviations from Anderson–Kim behaviour. HTS materials show giant flux creep effects.
Deviations from the straightforward logarithmic decay rate predicted by Anderson–Kim have
been reported extensively using a variety of standard and novel techniques. Initial reports
of non-Anderson–Kim behaviour came from long-time relaxation measurements. See for
example Yeshurun and Malozemoff 1988, Malozemoff (1991), Thompson et al (1991a) and
Sengupta et al (1993). Other types of measurement confirmed these observations using
for example flux creep annealing (Thompson et al 1991b, Sun et al 1990, Maley et al
1990), short-time relaxation (Gao et al 1992), and dynamic relaxation (Pust et al 1990,
Zhukov et al 1995, Perkins et al 1996) to name but a few. Zeldov et al (1990) proposed a
logarithmic form for the current dependence of the activation energy which has frequently
been observed. In order to explain deviations from Anderson–Kim behaviour, Hagen and
Griessen (1989), suggested that a distribution of activation energies should be taken into
account. In quite general terms, both the vortex glass (Feigel’man 1989) and collective
pinning theories describe a form for Ueff (J ) which diverges as J → 0. Feigel’man suggested
1602
L F Cohen and H J Jensen
an interpolation formula between the high-current Anderson–Kim limit and the low-current
regime which is frequently used, such that
µ
Jc
Uc
−1
(3.10)
U (J ) =
µ
J
where µ = (d + 2ς − 2)/(2 − ς ) > 0, d is the dimensionality of the relevant vortex
bundle volume and ς is the wandering exponent determined by equating the energy of an
elastic deformation to the fluctuation in pinning energy. Depending on the dimension etc, µ
can take various values for glassy behaviour, see Feigel’man (1989) and for the collective
pinning behaviour of an elastic vortex medium, see Blatter et al (1994b).
3.4. Magnetic measurement analysis
The description of irreversible phenomena in the mixed state of HTS has to be based upon
a self-consistent relation between the current density J and the flux density B, taking into
account the nonlinear current dependence of the creep activation barrier. Several related
methods have been developed to treat this problem. Hagen and Griessen (1989) developed
a model making it possible to calculate a distribution of activation energies for flux motion
from magnetic relaxation data using an exact inversion scheme. See also further work by
Griessen (1990, 1991).
3.4.1. The Maley method. The Maley method was introduced in 1990 (Maley et al 1990).
The basic idea was that one could extract Ueff (J ) directly from the time dependence of M.
Based on the general nonlinear form for Ueff (J ) from Beasley et al (1969), Maley et al
(1990) wrote down an expression for Ueff (J ) such that
Bdω
(3.11)
Ueff (J ) = −kB T ln |dM/dt| + kB T ln
τπ
where kB is the Boltzmann’s constant, τ is the thickness of the sample, d is the flux bundle
hopping distance and ω is the attempt frequency (see Sengupta et al 1993). In this approach
one first calculates T ln |dM/dt| at a given field. Then the data can be directly plotted
as Ueff versus (M − Mequ ), at different temperatures. Finally, by adjusting the constant
C = ln(Bωd/τ π) all of the data can be made to fall on the same smooth curve and the
Ueff − J relationship is obtained. The Maley method assumes that the characteristic current
and energy scales are temperature independent. A very useful example of the application
of the method is given in Sengupta et al (1993). Maley observed a logarithmic functional
form for Ueff (J ) in YBCO agreeing with the earlier Zeldov et al (1990) observation.
3.4.2. The generalized inversion scheme. Schnack et al (1993) introduced the generalized
inversion scheme which separates the effective activation energy into an energy term and a
current density term so that the rate equation can be written
Ueff (J, B, T ) = U0 (B, T )F [J /J0 (B, T )] = kT ln(Bdω/E)
(3.12)
where U0 is an energy scale, J0 (B, T ) a current density scale and F is a function
which describes the J dependence of Ueff (J, B, T ). Within this definition U0 and J0 are
closely associated with the pinning mechanism. The Schnack method is model dependent
because it requires that Ueff (T ) = [J0 (T )]p . Using this assumption, both Ueff (J, T , B)
and the parameter C = (ln(Bdω/E) can be directly extracted from relaxation and current
measurements.
Magnetic behaviour of superconductors
1603
3.4.3. The magnetic scaling analysis. The concept of magnetic scaling originates from the
observation that M–H loops at different temperatures can be brought to lie on top of each
other, producing a unique curve, if both the M- and H - axis are normalized appropriately.
This reflects one dominant physical process determining the behaviour in the temperature
regime of interest. Magnetic scaling has been reported frequently. See for example Zhukov
et al (1993), Kobayashi et al (1993), Oussena et al (1993) or Klein et al (1994). Scaling
relationships are also observed for the creep rate (Zhukov 1994) and the pinning force
density, J B (Civale et al 1991a).
The magnetic scaling analysis, introduced by Perkins (1995), builds on the Schnack et
al (1993) formalism and further considers the geometrical restrictions of moving through
the four-dimensional parameter E–J –T –B space. No assumptions are made about the form
of Ueff (J ) but the existence of magnetic scaling has certain implications.
Incorporating equation (3.12) into the rate equation and under the condition that M(H )
exhibits scaling, it can be shown that U0 (B, T ) and J0 (B, T ) must take the following forms:
U0 (B, T ) = 9(T )B n
(3.13)
J0 (B, T ) = λ(T )B m .
(3.14)
Differentiation of the rate equation with respect to ln B and ln E leads to the
relationship between χln = d ln J /d ln B, and the dynamic normalized creep rate S(B) =
[(ln J )/(ln E)]B,T at constant T :
χln = m + (nC − 1)Q
(3.15)
where C = ln(Bωd/E), and Q is the dynamic creep rate. Both χln and Q can be taken
directly from magnetization data.
3.5. Critical scaling applied to E–J curves
It is believed that in very pure crystals the transition between vortex solid and liquid will
be sharp and first order. In the presence of static disorder, it is further believed that this
transition broadens and become second order or continuous. If the transition is continuous
then the general rules which apply to all critical behaviour can be applied. Fisher et al
(1985) gave a general formulation of the scaling at and near a continuous transition. The
basic idea is that physical quantities near the transition can be expressed in terms of the
appropriate powers of a diverging coherence length ξ and coherence time τ , such that as
the transition temperature Tg is approached from above
ξ ≈ (T − Tg )−ν
and
τ ∝ ξ z.
(3.16)
A current scale Jsc for linear ohmic response is then defined as
Jsc = ckB T /φ0 ξ 2
(3.17)
here c denotes the speed of light, φ0 is the flux quantum, and kB is Boltzmann’s constant.
A non-ohmic response at all current scales is a signature of a critical transition. The E–J
characteristic for a d-dimensional sample is predicted to be a power law of the form
ln E ∝ [(z + 1)/(d − 1)] ln J.
(3.18)
For T greater than Tg the characteristic changes from ohmic behaviour for small current
densities where
ρ(T ) ∝ (T − Tg )ν(z+2−d)
= (T − Tg )s
(3.19)
1604
L F Cohen and H J Jensen
to power law at large current densities.
For T less than Tg the E–J characteristic is always nonlinear and for small current
predicts a resistivity given by
ρ = E/J = ρ0 exp[−(Jc /J )µ ]
(3.20)
this implies a negative curvature on a ln E–ln J plot. At large current densities it is predicted
to again have critical power law behaviour. The crossover current vanishes as (T −Tg )ν(d−1) .
These scaling ideas can be applied to any continuous transition. An example is the
Kosterlitz–Thouless transition appropriate to two-dimensional thin films where d = z = 2
and E = J 3 (Hebard and Fiory 1982). These scaling laws have been applied to the E–J
curves in HTS and the values of the exponent ν, the dimension d and the dynamic exponent
z, have been used extensively as evidence for the existence of melting and a transition to a
vortex glass phase. The validity of these claims are reviewed in more detail in section 4.
Alternative scenarios exist, see for example Kiss and Yamafuji (1996).
For T less than Tg critical scaling is not observable at high driving forces. The influence
of disorder on the vortex system as it freezes is strongly tuned by the magnitude of the drive
current in a manner not addressed by scaling theory. Koshelev and Vinokur (1994) and
Koshelev (1996b) investigated this theoretically. They predicted that in the presence of static
disorder, freezing will take place into a perfect lattice if the lattice is moving sufficiently fast
and that plastic or glassy behaviour will be observed otherwise, depending on the magnitude
of the drive current with respect to Jc . In this scenario, a current-driven transition into a
perfect lattice can be visualized as occurring well below the freezing transition.
4. Experimental observation of vortex behaviour
4.1. Reversible properties
4.1.1. Hc1 . Measurement of Hc1 is usually made by examining the point of departure from
linearity on the initial slope of the magnetization curve. Without pinning there is of course
a sharp cusp at Hc1 . In HTS this measurement is difficult because pinning causes only
a subtle departure from linearity and deviation from the M = −H is small at Hc1 . For
H //c, the sample geometry (which is usually plate-like) and self-field effects contribute to
the shape of the initial slope. Surface barriers to flux penetration result in enhanced values
of Hc1 and measurements yield an upper bound only (Umezawa et al 1988). Microwave
measurements of the change in the penetration depth as a function of DC magnetic field,
yield clean data of the first effective field at which vortices penetrate single crystals (Wu
(0) = 200 Oe.
and Sridhar 1990). For YBCO, Hcc1 (0) = 800 Oe, Hcab
1
Not surprisingly perhaps, given these complications, Hc1 has sometimes been reported
) geometry,
not to follow the expected GL form for either the H //c(Hcc1 ) or H //ab(Hcab
1
particularly for (T /Tc ) 6 0.5 (Krasnov et al 1991). It has been reported that within a
few Kelvin of Tc , Hcc1 disappears. In YBCO crystals this is interpreted as a thermally
induced excitation over the surface barrier (Safar et al 1990, Pastoriza et al 1994b). The
situation for BSCCO 2212 from Brawner et al (1993b), is illustrated in figure 7. Brawner
et al interpreted the disappearance the penetration field Hp (which can be considered as an
upper bound for Hc1 ) as evidence for thermally induced vortex–antivortex pairs (Kosterlitz–
Thouless-type transition) or alternatively as being related to order parameter fluctuations
within vortices. (Blatter et al 1993).
When the field is aligned within 6◦ from the ab planes, a sharp change in the initial
magnetization in BSCCO has been observed by Nakamura et al (1993). They interpret this
Magnetic behaviour of superconductors
1605
Figure 7. (a) The temperature dependence of the critical current density Jc with an inset
showing the field-cooled (5 Oe) magnetization of the crystal. Also the penetration field Hp
versus T with the inset showing a suggested phase diagram, both for a BSCCO 2212 crystal
with H //c. Below 83.5 K, Hp extrapolates linearly to Tc = 86 K. Hp is absent above 83.5 K.
(b) Shows Jc and Hp for a YBCO 123 crystal. Both persist to Tc in this case. From Brawner
et al (1993b).
as an effective three-dimensional to two-dimensional transition in the shielding current path
because the c-axis coherence is lost across the thickness of the crystal. They attribute this
to the fact that the Josephson current is suppressed by the entrance of flux between the ab
in TlBaCuO-2201 (Hussey et al
planes. Similar observations have been reported for Hcab
1
1994, 1996). Changes in the reversible screening current path for H //ab and T close to
Tc have not been reported for YBCO crystals.
The values of Hc1 in the two field orientations can be used to determine the GL
superconducting mass anisotropy ratio γ . The problem of internal sample flatness and
external sample alignment when the field is aligned parallel to the sample surface, throws
doubt on the precision of many measurements. Nevertheless some consensus has been
lies between 4 and 7 for well oxygenated YBCO and between 30
reached. Hcc1 / Hcab
1
and 300 for BSCCO 2212 (Martinez et al 1992). λab and λc are determined from the
measurement of Hc1 , from AC susceptibility and from microwave measurement. Typical
values are λab = 140 nm, λc = 600 nm for YBCO and λab = 185 nm, λc = 7.5 µm for
BSCCO 2212.
4.1.2. Hc2 . The mean-field approximation for Hc2 is not valid in the presence of large
thermal fluctuations. Only specific heat measurements will determine unequivocally whether
there is a first- or second-order transition at Hc2 . For LTS materials, Hc2 (T ) acts as a field
scale for all elastic properties of the vortex system. Whether Hc2 (T ) or some other field
line such as the melting line plays this role in HTS vortex behaviour is still unclarified.
Two non-calorimetric methods are used to determine Hc2 . A line in the H –T plane
has been extracted from resistivity measurements (Mackenzie et al 1994, Smith et al 1994,
1606
L F Cohen and H J Jensen
Osofsky et al 1993, 1994) which place a lower bound on Hc2 by identification of the field at
which the resistivity reaches 90% of its extrapolated normal state value at that temperature.
These measurements either require huge magnetic fields to be applied of the order of 140 T,
or HTS samples which have been doped in such a way that their Tc and Hc2 values are
greatly depressed. (Note that such manipulation of the charge carrier state in the material
either by ‘overdoping or underdoping’ can change normal state properties, by opening up a
pseudogap in the normal density of states. The pseudogap has been studied by specific heat
capacity, NMR, inelastic neutron scattering and thermo-electric power. The anisotropy of
the superconducting properties are also modified by doping.) The form of the line extracted
from the resistivity curves is similar to the irreversibility line and indeed probably reflects
the irreversibility line rather than the Hc2 . These resistive measurements are also hampered
by the large broadening of the transition in a magnetic field due to flux flow. Even as a
very approximate lower bound it is nevertheless clear that Hc2 (T = 0) must indeed be very
large.
Another procedure used to estimate Hc2 is to find the temperature at which the reversible
magnetization approaches zero for a given field. This can be a problem for HTS materials
because many samples have rare earth ions which are paramagnetic. Hc2 (T = 0) can been
estimated from the reversible magnetization close to Tc (Welp et al 1989), using either a
linear analysis (see figure 8) or a more complex procedure (Hao and Clem 1991). According
to the theory of Werthamer et al (1966),
δHc2
.
(4.1)
Hc2 (0) = 0.71Tc
δT Tc
This gives Hc2 (T = 0) = 140 T for well oxygenated YBCO 123. There are problems
in using this method for BSCCO, because of broadening of the transition in magnetic field.
is used to estimate γ of the order of 5 in YBCO 123 and 30 in BSCCO
The ratio Hcc2 /Hcab
2
2212. Measuring Hc2 = φ0 /2πξ 2 yields ξ . In YBa2 Cu3 O7 , for H //c, ξab = 15.4 Å.
As oxygen is removed ξab remains at this value initially and then it increases significantly
(Ossandon et al 1992, Zhukov et al 1994). M(T , H ) is at least an order of magnitude
smaller when H //ab and therefore Tc (H ) is rather ill-defined. This means that even in
the least anisotropic material, measurements of Hcab
(0) are not particularly accurate. For
2
well oxygenated YBCO, ξc = 3–5 Å. This value is to be compared with the distance
between the superconducting layers which in YBCO 123 is 8 Å or the total unit cell length
of 11.7 Å. It is thought that because the Cu–O chains of fully oxygenated crystals are also
superconducting (Kresin and Wolf 1992), and it is the chain–plane separation 4 Å which
should be compared with ξc = 3–5 Å.
In BSCCO 2212 ξab = 10 Å implying from the measurement of γ that the value of ξc is
much smaller than the interplanar spacing. In this case the GL anisotropic three-dimensional
model breaks down and must be replaced by a quasi-two-dimensional model (Lawrence and
Doniach 1971).
4.1.3.
Influence of columnar defects. Thermal fluctuation effects are particularly
pronounced in the most anisotropic materials. Kes et al (1991) measured a weak fluctuation
contribution to the reversible magnetization up to 20 K above the field-dependent transition
temperature Tc (B). The reversible magnetization at different magnetic fields applied along
the c-axis, becomes field independent at some temperature T ∗ < T ∗ , where T ∗ is known
as the ‘crossing point’ (Kes et al 1991, Tesanovic and Xiang 1991). Below T ∗ , vortex
positional fluctuations (phase fluctuations) modify the field dependence of the magnetization
and suppress the pinning critical current density (Feigel’man and Vinokur 1990). At T ∗ the
Magnetic behaviour of superconductors
1607
Figure 8. (a) Temperature dependence of the reversible magnetization in YBCO in magnetic
fields of 50 gauss and 5 T with the field applied parallel to the planes. (b) Temperature
dependence of the magnetically determined upper critical field. The slopes of linear extraction
are indicated. The broken lines represent the upper critical points taken from resistive zero
points. After Welp et al (1989).
magnetization becomes field independent because the logarithmic field dependence of the
mean-field magnetization is cancelled by the same logathimic dependence in the entropy
contributions of pancakes decoupled along the c-axis (Bulaevskii et al 1992, Tes̆anović et al
1992). The observation of the crossing point is theoretically considered to be the strongest
support for the discrete nature of the pancake vortices.
In columnar defected BSCCO 2212 crystals this behaviour is modified, see Bulaevskii et
al (1996). A maximum is observed in M and the crossing point phenomenon is suppressed
1608
L F Cohen and H J Jensen
for fields 0.2Bφ 6 µ0 H 6 Bφ . Bφ = nφ φ0 is known as the matching field associated
with a certain density of columnar tracks, nφ . See van der Beek et al (1996), Qiang Li et
al (1996) and Pradhan et al (1996). It is still unclear exactly what role columnar defects
play and whether they effectively enhance coupling between layers and correlations along
the c-direction (Bulaevskii et al 1996). The absence of the crossing point phenomenon
suggests that the random distribution of columnar defects suppresses the interaction between
fluctuations in the critical regime (van der Beek 1996). The crucial message at this point is
that it is clear that vortex pinning not only affects irreversible magnetic properties but can
also affect its thermodynamic properties.
4.2. The irreversibility line
In 1988, Yeshurun and Malozemoff had already recognized the existence of giant flux creep
in HTS. It was quickly established that a line existed well below Hc2 that separated reversible
from irreversible magnetic behaviour. Below it, because irreversible magnetization was
observed, vortices were thought to be pinned and this pinning was associated with a vortex
solid. Above the line, the vortices were considered to be unpinned and under the influence
of a driving force—flux flow or a liquid state existed. As we discussed in section 2,
these initial conclusions have since been re-examined because pinned liquids might exist
below the irreversibility line and an unpinned perfect Abrikosov lattice might exist above
it. Over the past eight years there has been an active discussion as to whether the Hirr line
represents simple depinning or whether it is coincident with melting or decoupling. Cooper
et al (1997) presented experimental evidence which appears to show that thermodynamic
fluctuations play an important role in determining the irreversibility line and reversible
magnetizatisation in YBCO 123 crystals. Cooper et al (1997) found that the form of
Hirr (T ) is consistent with the three-dimensional XY model which is based on fluctuations
of the order parameter.
As vortices depin (or become pinned), dissipation associated with their movement
generates an increase (decrease) in voltage. The change in the vortex properties at Hirr
may reflect the existence of a phase transition. However, the disappearance (or appearance)
of an irreversible signal is intrinsically non-equilibrium and depends on experimental method
and criteria. For clarity, we discuss the evidence for melting and decoupling in separate
sections.
4.2.1. The position of the Hirr line in the H –T plane. Optimally doped rare earth (Re)BCO
123 and BSCCO 2212 crystals have similar Tc values and yet the position of their Hirr lines
in the H –T plane are strikingly different, as shown in figure 9. Assuming that Tc reflects
the strength of the in-plane superconductivity, then the Hirr line is clearly affected by other
factors such as the superconducting anisotropy ratio γ , and the c-axis coupling strength.
At T /Tc = 0.75, Tallon (1994) found that the position of the Hirr line is exponentially
dependent on the inter-plane or plane–chain separation for many families of optimally doped
materials as indicated in figure 10. The exponential relationship strongly suggests a weak
link plane–plane or plane–chain coupling mechanism. (Implying that the chain is also
superconducting). The Tallon group have also shown that by careful cation substitution and
oxygen doping, Tc can be kept constant while the c-axis coupling strength is weakened. In
these samples Tc is constant but Hirr (T /Tc = 0.75) falls, strongly supporting the premise
that this is the main difference between the YBCO and BSCCO Hirr lines.
The position of the Hirr line in the H –T plane, also depends on the vortex and defect
dimensionality and the defect density. Columnar defects produced by heavy ion irradiation,
Magnetic behaviour of superconductors
1609
Figure 9. The irreversibility field versus reduced temperature T /Tc for a series of deoxygenated
TmBaCuO7 d (123) crystals (+) with increasing oxygen from left to right across the graph, a
YBaCuO8 (124) sample () and a BiSrCaCuO 2212 crystal (*), using a criteria of 10 A m−1
cut-off in the closing of the M–H -loops. After Cohen et al (1994b).
enhance the Hirr line, as illustrated in figure 11, for the relatively anisotropic Tl-2223 and
Tl-1223 systems. It is energetically favourable for pancake vortices to line up along the
columnar track, and by doing so phase fluctuations between planes are suppressed. In this
way, although the coupling between the planes is not necessarily altered physically, the
system can mimic the vortex behaviour in more three-dimensional systems.
In YBCO 123, unidirectional enhancement of the Hirr line along the direction of the
columnar was interpreted as a reflection of the line nature of the pre-existing vortex structure
(Civale et al 1991a, b). When 200 nm thick YBCO films were irradiated with 5 GeV Pb
ions to produce columnar defects, both the screening current and the Hirr line were enhanced
preferentially when the vortices were aligned parallel to the columns (Prozorov et al 1994,
Fischer 1992). BSCCO 2212 crystals have significant uniaxial enhancement of Hirr due to
columns above 40 K as shown by Klein et al (1994, 1993b), and Thompson et al (1992).
This is illustrated in figure 12(a). It is generally accepted that below 40 K the vortices
become more two-dimensional-like so that isotropic point-defect pinning becomes more
favourable. However, columnar defects are thought to suppress phase fluctuations and it is
possible that at lower temperatures this is simply no longer significant. The Hirr line and
the screening current density J are greatly enhanced over the unirradiated crystal down to
15 K (Klein 1993b). Below 15 K the pre-existing point defects are more effective than the
columns at pinning defects. Zech et al (1995) have demonstrated that the unidirectional
enhancement also disappears within 10 K of Tc , suggesting that thermal fluctuations
eventually reduce the usefulness of the columns. Note that columnar defects influence
the behaviour of phase fluctuations as reflected by reversible magnetization properties at
temperatures approaching Tc . Gray et al (1996, 1997) discussed the contradition of three-
1610
L F Cohen and H J Jensen
Figure 10. (a) Temperature dependence of the irreversibility field H ∗ for various materials. (b)
H ∗ at T /Tc = 0.75 versus separation of CuO2 planes (di ) for various materials.
Magnetic behaviour of superconductors
1611
Figure 11. The Hirr line enhancement resulting from columnar defects of different densities, in
Tl-2223 and Tl-1223 systems. After Brandstätter et al (1995).
dimensional-like uniaxial enhancement due to columns at temperatures within the twodimensional thermally activated limit (see sections 4.1.3, 4.3.1 and 4.4.2).
The interplay between different types of defects which may be present in a crystal
such as point defects (e.g. oxygen vacancies) and planar defects (twinning planes) makes
it impossible to summarize all possible behaviours. Twin boundaries are reported to act as
strong pinning sites in some situations. However, Oussena et al (1995, 1996) reported that
in unidirectional microtwinned crystals at low magnetic fields the twin boundaries act as
vortex channels. Figure 13 illustrates the case highlighted by Jahn et al (1995) who found
that in YBCO 123, point-like defects can depress the Hirr line without significantly lowering
Tc , whereas correlated disorder such as intrinsic pinning of the planes or twin boundaries
enhance the Hirr but lower the screening current density. Recently, Flippen et al (1995)
showed that even surface damage can significantly alter the pinning properties of YBCO
single crystals which have very low twin boundary densities.
4.2.2. The temperature dependence of the Hirr line (H //c geometry). Almasan et al (1992)
suggested that the Hirr (T ∗ ) line obeyed a scaling relationship universal for all HTS, whereby
at temperatures above T ∗ /Tc > 0.6 the Hirr (T ∗ ) exhibited a power law dependence of the
form
Hirr (T ∗ ) = (1 − T ∗ /Tc )m
(4.2)
where m = 32 and T ∗ (H ) is the temperature at which the resistance R(H, T ) drops to 50%
of its normal state value, at a fixed field, as shown in figure 14. At lower temperatures a
crossover to a more rapid dependence occurs as shown in figure 14 (see the break away
on the right-hand side of the figure). Although such scaling exists, it is not as general
as first implied by Almasan et al (1992). For example the Hirr line of optimally doped
YBCO 123 shows no upturn down to the lowest temperatures measured. The Hirr (T ) of
BSCCO 2212 is illustrated in figure 15, after Schilling et al (1993). It follows a power
law form with an exponent m = 2 at high temperatures and a much stronger upturn at low
1612
L F Cohen and H J Jensen
Figure 12. (a) Magnetization curves at 40 K and 60 K for a BSCCO 2212 crystal which has
been irradiated with Pb ions at 45◦ to the c-axis. The field is applied at +45◦ and −45◦ to the
c-axis. After Klein (1993b). (b) The effective enhancement of the Hirr line in BSCCO 2212
crystals for continuous columnar defects produced from Pb ions which are effective below about
60 K and Xe ions which create cluster defects.
temperatures than found in the intermediately anisotropic materials. The significance of the
low temperature upturn in Hirr , has been interpreted as a softening of the elastic parameters
of the vortex system making the existing pinning sites more effective and enhancing the
Hirr line. Schilling related this softening to a dimensional crossover in the vortices from
three-dimensional at higher temperatures to two-dimensional at lower temperatures.
The Hirr line as measured by torque magnetometry (refer to Farrell 1994) coincides with
sharp features in the resistive transition as measured by Safar et al (1992c), in untwinned
YBCO crystals with very low point-defect density. It appears that in this case Hirr coincides
with the melting line, i.e. that pinning disappears at the solid-to-liquid transition. Once static
disorder is introduced by irradiation or the underlying static disorder becomes effective
because the system is cooled, the melting line and the bulk-depinning line separate.
This is also consistent with the predictions of the three-dimensional XY model as
discussed in a brief review by Cooper et al (1997). The three-dimensional XY model
Magnetic behaviour of superconductors
1613
Figure 12. (Continued).
is based on fluctuations of the order parameter, and predicts a melting line of the form
described in equation (4.4) with α = 1.3. Cooper et al found that both the Hirr line and
the reversible magnetization scale according to the predictions of the three-dimensional XY
model. Several groups have shown that the electronic specific heat (Overend et al 1996
and references therein), magnetization (Hubbard et al 1996, Liang et al 1996) and electrical
resistivity (Howson et al 1995) obey the three-dimensional XY scaling laws associated with
critical thermodynamic fluctuations.
However, the coincidence of melting and depinning is strongly disputed for BSCCO
crystals (Farrell et al 1995, 1996). Zeldov et al (1994) have shown that in perpendicular
applied magnetic fields, vortex penetration is delayed significantly in disk-shaped samples
due to the presence of a potential barrier of geometric origin. The geometric barrier produces
hysteretic magnetization in the absence of bulk pinning. Majer et al (1995) used local Hall
probe measurements to show that when a BSCCO 2212 crystal is shaped so that its surface
gradient is greater than that of the ellipse making up its extremal dimensions then the Hirr
line due to the geometric barrier is reduced to zero. This is illustrated in figure 16. The
jump in magnetization marked in the figure by Hm is associated with the melting line.
This study suggests that in BSCCO 2212 an Hirr line (although not the one associated
with bulk pinning) can be positioned quite separately on the H –T plane to the melting line.
However, the topic is controversial and may only be resolved when there is an improvement
in experimental sensitivity.
4.2.3. The Hirr line as a function of magnetic field orientation. Remembering that the G–
L parameters are anisotropic, one can nevertheless learn about vortex dimensionality and
the influence of defect dimensionality on vortex behaviour, by varying the magnetic field
orientation with respect to the ab planes. In transport measurements the magnetic field
and current directions can be controlled separately. In a magnetization measurement this
is not the case and the screening current is induced in a direction either perpendicular to
1614
L F Cohen and H J Jensen
Figure 13. Magnetization loops for two YBCO crystals at 77 K. W1 is a twinned crystal with
very low density of point defects and W2 is doped with Sr and has a high density of point
defects. W2 shows a large J but a depressed Hirr line compared to W1. After Jahn et al (1995).
the applied field or in a direction controlled by sample or defect geometry. The induced
screening current should always be perpendicular to the field, a field tilted away from the
c-axis will produce two components of current (Gyorgy et al 1989). However, geometry
effects in flat samples (Zhukov et al 1997) and linear or planar defects (Küpfer et al
1996), can prevent the magnetic moment from tracking the applied field. Making magnetic
measurements in this case requires more care.
When the field is close to the ab planes, quantitative analysis is difficult because the
vortex system may take a new structure related to the extreme angle. For example, in YBCO
crystals at low fields and angles close to the ab planes, high resolution Bitter patterns reveal
that the flux-line lattice which lies in the tilt plane takes the form of vortex chains, as a result
of attraction within the tilt plane (Gammel et al 1992, Grigorieva et al 1993). In BSCCO
similar attractive forces results in the coexistence of two vortex species, oriented parallel to
the ab planes and the c-axis (Grigorieva et al 1995). When the field is aligned accurately in
the planes, a lock-in transition occurs and has been observed in untwinned YBCO crystals
ab
(T ) occurs at around 80 K which
(Ossandon et al 1992). A crossover in the form of Hirr
has been associated with the disappearance of the coherence in the c-axis current (Lacey et
al 1994, Ossandon et al 1992). Such a crossover is predicted to occur when the thermal
energy washes out the inter-plane coupling (Fischer)- formulae.
The variation of Hirr (T ) with θ, the angle between H and the ab plane, can be used to
determine the dimensionality of the vortex structure at different temperatures. For example,
Kes et al (1990) proposed, that the field component parallel to the c-axis, H sin θ determines
Magnetic behaviour of superconductors
1615
Figure 14. Log(H /H + ) versus log(1 − T ∗ /Tc ), where H + is a normalizing field and T ∗ (H )
is the temperature at which the normal state resistance falls to 50% of its normal state value.
The full curves in all four figures are power-law behaviour with the exponent m = 32 . (e) is a
composite plot of all the data from figures 2(a)–(d ). From Almasan et al (1992).
Hirr for quasi-two-dimensional pancake vortices. Various angular scaling relationships have
been suggested: Tachiki and Takahashi (1989), Klemm and Clem (1980), Tinkham (1963).
These scaling models breakdown in a variety of ways when line-like and planar defects
dominate the pinning behaviour or when Josephson kink structures are created at angles
close to the planes (Zhukov et al 1996). In general, anisotropic G–L theory suggests that
H = HA (sin θ 2 + (1/γ 2 ) cos θ 2 )−1/2 .
(4.3)
Once γ > 15 this expression is, for small θ indistinguishable from Kes scaling (Iye et
al 1992). In YBCO crystals containing a homogeneous point-defect distribution, the Hirr
line moves systematically upwards in the H –T plane as the field is swung away from the
c-axis. It is found to be about five times higher when it lies approximately in the plane, in
agreement with equation (4.3) and measured values of γ (see Angadi et al 1991).
1616
L F Cohen and H J Jensen
Figure 15. The Hirr line, interpreted as a melting line, is plotted on a logarithmic field scale for
BSCCO 2212 crystals. The arrows indicate the temperature range where the data has been fitted
to expressions for two-dimensional (low temperature) and three-dimensional melting lines. The
full curve corresponds to a description in terms of a Josephson coupled layered superconductor
(JCLS). The broken curve is a quadratic fit to the data near Tc . After Schilling et al (1993).
Schmitt et al (1991) showed that for BSCCO films measured in the maximum
Lorentz force geometry (H ⊥ J ), the maximum critical current in the planes is obtained
when H //ab and it is field independent due to strong intrinsic pinning by the planes.
Equation (4.3) does not describe the behaviour well because γ reflects the mass anisotropy
of the superconducting electrons and not the strongly anisotropic pinning and flux creep
effects found in BSCCO (see Kobayashi et al 1995a, b). The difference between the pinning
(or current) anisotropy and γ has been explored in YBCO crystals as a function of oxygen
by Thomas et al (1996).
Nelson and Vinokur (1993) predicted that correlated disorder such as columnar defects,
twin or intrinsic planes, can transform the vortex solid into a Bose glass. The Bose glass
is predicted to have a distinctive Tirr (θ) cusp-like dependence when the field component
along the disorder is fixed, and is less than or equal to the matching field (one vortex per
column). The predicted behaviour is shown in figure 18. Tirr increases with increasing ab
plane field component (increasing tilt) for a vortex glass and decreases with increasing tilt
for a Bose glass. The behaviour of Tirr can be explained more simply. In the unirradiated
crystals a simple vector model argument along the lines of equation (4.3) can be used. In
the irradiated crystals, the reduced effectiveness of the columns to pin the vortices as the
field is tilted away from the line-like defect, would produce the cusp-like feature in Tirr .
The cusp-like feature has been observed in Tirr in columnar-defected YBCO crystals (Jiang
et al 1994, Reed et al 1995), intrinsic planes of YBCO (Kwok et al (1992) as shown
in figure 17(b)) and columnar-defected BSCCO (Seow et al 1996) crystals. The latter is
illustrated in figure 28. The enhanced Tirr (θ) is only observed at high temperatures. Zech
et al (1995) have shown how Kes scaling of the Hirr line breaks down in irradiated BSCCO
Magnetic behaviour of superconductors
1617
Figure 16. Local magnetization loops B–Ha versus Ha in BSCCO crystals of (a) platelet and
(b) prism shapes at T = 80 K. The platelet crystal shows hysteretic magnetization below the
irreversibility field HIL . In the prism sample the geometric barrier is eliminated and a fully
reversible magnetization is obtained at temperatures above 76 K. After Majer et al (1995).
crystals as the field is brought into line with the columnar tracks.
Küpfer et al (1996) clarified the relationship between the variation of the screening
current density J (T , θ, B) and the Hirr (θ, T ) as a function of angle in a series of YBCO
crystals with varying degrees of random point-like disorder (vortex glass-like) and twin plane
disorder (Bose glass-like). In the case of flux channelling along twin planes, minimum
pinning and maximal Hirr line is produced when the field is aligned to the c-axis, as
illustrated in figure 19. Considerably more complex behaviour can result when crystals
have different types of disorder.
1618
L F Cohen and H J Jensen
Figure 17. (a) The transport current of epitaxial BSCCO 2212 films as a function of the angle
between the magnetic field and the c-axis. There is a maximum in the critical current when
H //ab is field independent. After Schmitt et al (1991). (b) The angular dependence of the
vortex lattice melting temperature Tm (θ ) of a twinned YBCO crystal. The inset shows the plot
of Tm versus the angle displaying the cusp near H //c. The line is a guide to the eye. After
Kwok et al (1992).
4.3. In the vicinity of Hirr
If the Hirr line coincides with a well defined melting line Hm , the vortex system will be
a pinned solid below the line and an unpinned liquid above it. If melting and decoupling
occur simultaneously with the Hirr line the pinned solid will transform above the line into
a two-dimensional lattice or gas (as determined by the density of vortices).
These pictures are straightforward to visualize but experimental differentiation between
them is not easy. How does one set about doing this? At the transition between an unpinned
vortex lattice and an unpinned vortex liquid the ab plane resistivity ρab should register a
change in shear viscocity (see section 4.5.1). Within each regime ρab is ohmic, with a value
which is some fraction of the normal state resistivity. More usefully perhaps, the c-axis
Magnetic behaviour of superconductors
1619
Figure 17. (Continued)
correlation (or line integrity) is lost in the liquid state when the vortices are entangled as
a result of thermal disorder (Nelson 1988). Hence, a non-Lorentz geometry with J and
H //c, ρc will be finite in the vortex liquid (where phase slip processes associated with
vortex cutting and rejoining will create dissipation). For a three-dimensional vortex lattice
ρc = 0. A quasi-two-dimensional vortex lattice also has only weak c-axis correlation.
Differentiating between a quasi-two-dimensional solid and an entangled three-dimensional
liquid, is yet more subtle.
4.3.1. The disentangled vortex liquid and the c-axis correlation. Fendrich et al (1995)
examined the question of pinning in the liquid state by looking at the effect of pointdefect electron irradiated on clean untwinned YBCO crystals. In the unirradiated crystals
a linear free flux flow resistivity was found to follow the simple Bardeen Stephen model
ρff = (B/φ0 Bc2 )ρn , where ρn is the normal state resistivity. In the irradiated crystals, the
linear resistivity was depressed relative to the unirradiated at the same reduced temperature,
below a certain temperature denoted Tp . As the defects in this sytem were uncorrelated,
individual vortex pinning was found to be too small to explain the decrease of ρ in the postirradiated liquid. The authors attributed the depression of ρ to an increase in viscosity as
a result of increased liquid entanglement in the irradiated crystals. They took into account
the viscous shear processes that take place in liquid flow and related this to a plastic
energy proportional to the energy involved in vortex cutting and recombination. They also
found that the resistive transition was broadened as a function of the point disorder (see
section 4.5.1) and the Hirr line was depressed.
Correlated disorder will disentangle, localize (or pin) the vortex liquid as claimed by
Civale et al (1991a, b) in the case of columnar defects, and by Flesher et al (1993), in the
case of twin boundaries. The correlation along the c-axis or degree of entanglement can be
determined by measuring ρc directly. A pseudoflux transformer set up along the lines of
Giever (1965), measures vortex line tension which also reflects the strength of the c-axis
correlation.
In the pseudoflux transformer configuration (figure 21), eight electrical contacts are
attached to the crystal. A driving current is made to enter and leave the sample through
leads at the top surface of the sample and external magnetic field is applied perpendicular
1620
L F Cohen and H J Jensen
Figure 18. (a) The phase diagram in the (T , Hperp ) plane with the field Hz along the direction of
the fixed correlated disorder. The crystalline phase is an Abrikosov lattice for fields tipped away
from a single family of parallel twins or it represents a smectic-like phase for columnar pins or
a mosaic of twin boundaries. (b) The phase diagram in the limit of strong correlated disorder.
Interactions are important in determining the localization length and transport at intermediate
current scales above the broken crossover curve B ∗ . After Nelson and Vinokur (1993). (c) The
resistively determined irreversibility line Tirr before (BI) and after irradiation (AI). Also shown
is the Bose glass temperature TBG . (d ) Angular dependence of Tirr in an irradiated crystal (lefthand axis) compared to an unirradiated crystal for fixed Bz parallel to the c-axis when the field
component parallel to the ab-planes Bperp is increased. After Seow et al (1996).
Magnetic behaviour of superconductors
1621
Figure 18. (Continued)
to the superconducting CuO planes. The voltage drop at the surface Vtop and at the bottom
Vbot of the sample is measured simultaneously. The system is studied in the linear I –V
region at T > Tirr as well as in the nonlinear I –V region. (The irreversibility line is defined
here simply as the temperature below which limJ →0 dE/dJ = 0.)
Lopez et al (1994a, b, 1996) used the DC flux-transformer configuration to compare
the behaviour in twinned and untwinned YBCO crystals, and there are quite remarkable
differences between the two. In the twinned crystals, Vtop 6= Vbot for temperatures T > Tth
(th for thermal cutting of the vortices). The dependence of Tth is studied as function of the
external magnetic field. The electric field induced by the motion of vortices are according
to the Josephson relation (see e.g. Tinkham 1995) given by E ∝ nv where n denotes the
density of the moving vortices and v is the velocity with which they move. Since the
same number of vortices are induced in the top of the sample as in the bottom, the authors
conclude that the velocity of the segments of the flux lines in the top of the sample must be
1622
L F Cohen and H J Jensen
Figure 19. Angular dependence of the normalized hysterisis width δm(φ)/δm(φ = 0) taken at
the peak position (open squares). Measurements are made at (a) 80 K for crystal number 1 and
77 K for all other crystals. The full squares represent the corresponding values of the normalized
irreversibility fields Birr . The crystal in (a) has strong twin-plane pinning, (b) and (c) have twin
planes and point defects, (d ) has point defects only. After Küpfer et al (1995).
different from those in the bottom when Vtop 6= Vbot . The current voltage curves (IVCs) are
linear down to temperatures Ti which is associated with the irreversibility line. Above Tth
the coherence across the thickness of the sample is lost and flux cutting must occur. Below
Tth the flux lines are coherent across the sample thickness. However, between Ti and Tth ,
at large enough currents, denoted Ic (the cutting current), Vtop 6= Vbot . Ic is dependent on
the magnetic field and thickness of the crystal.
Figure 22 shows a flux transformer in twinned and untwinned crystals. Twin planes
acting to disentangle vortex liquid. After Lopez et al (1996).
For twinned crystals, the region between Ti and Tth is thought to be a disentangled
vortex liquid, where the twin boundaries stabilize the disentanglement (i.e. they hold the
vortices straight), but only effectively up to a finite shear force associated with Ic . In the
untwinned crystal a sharp resistive transition is observed at a temperature Tm . Below Tm ,
Vtop = Vbot at all drive currents, and the IVCs are nonlinear for any current drive. Above
Tm , Vtop 6= Vbot and the IVCs are linear at all currents. In the untwinned crystal, the authors
attribute the behaviour above Tm , to an entangled vortex liquid and below Tm , to plastic
motion of a vortex solid which is correlated in the c-direction.
Interpretation of flux transformer data is complicated by the question of whether
non-local resistivity has to be taken into account. Safar et al (1994) claimed that in
heavily twinned YBCO, the line-like nature discussed above implies that a simple local
electrodynamic picture (that the local electric field is simply determined by the local current),
breaks down. However, Eltsev and Rapp (1994, 1995a, b) successfully explained their
Magnetic behaviour of superconductors
1623
Figure 20. (a) Shows the resistivity of an untwinned YBCO single crystal versus normalized
temperature, at 4 T for H //c. The sharp resistive transition at Tm occurs in the virgin crystal.
The transition broadens when the crystal was irradiated with 1 MeV electrons. (b) Shows
voltage–current characteristics before and after electron irradiation at several different reduced
temperatures T /Tc for H //c. After Kwok et al (1993).
1624
L F Cohen and H J Jensen
Figure 21. Sketch of the Giaever DC flux transformer and pseudo DC flux transformer contact
configuration. In the latter, due to the anisotropy in the resistivities, injecting current in the top
face produces an inhomogeneous current distribution. After de la Cruz et al (1994b).
Figure 22. Normalized resistance versis reduced temperature of twinned and untwinned single
crystals measured in a flux transformer geometry. After Lopez et al (1996) who argue that twin
planes act to disentangle vortex liquid.
Magnetic behaviour of superconductors
1625
transformer measurements in twinned YBCO by a local anisotropic superconductor. There
is a contradiction here. Interpretation of flux transformer data hinges on the temperature
dependencies of ρab and ρc . In the twinned YBCO crystals Safar et al (1994), found that ρc
goes to zero at Tth . (This data is only shown on a linear scale so results may be inconclusive.)
In a local picture, if ρc is finite, the current is distributed across the crystal leading to flux
cutting and Vtop 6= Vbot . If ρc = 0, then the same current runs along each ab plane and
one would expect that Vtop = Vbot . Those that believe that the local resistivity picture is
correct would say that twin planes reduce the thermally induced phase fluctuations, so that
ρc drops more rapidly in crystals with correlated defects.
BSCCO crystals, unlike YBCO are not naturally twinned, so that the only static disorder
is point disorder. The flux transformer data is much more straightforward to interpret (Safar
et al 1992a, Busch et al 1992, Wan et al 1993). The in-plane dissipation associated with an
ab plane electrical transport current is found not to be correlated over the sample thickness
over wide portions of the H –T plane. The flux transformer data shows that Vtop and Vbot are
never equal and in increasing field, the difference between them increases (see figure 23). It
is concluded that the vortices, must cut and reconnect during transport, suggestive of twodimensional vortices. A peak is seen in Vbot which is depressed in low fields. The peak is
probably associated with the temperature at which Josephson coupling becomes important.
Related to this, Fuchs et al (1996) showed that Vtop = Vbot at the melt temperature in
BSCCO.
Doyle et al (1996) made transport measurements in the flux transformer and c-axis
geometries in heavily columnar defected BSCCO crystals. They found that as in twinned
YBCO, there is a range of fields and temperatures where Vtop and Vbot show close
correspondence. They successfully described their transformer data using a local anisotropic
electrodynamics picture only and compared with directly measured ρc data. In the fieldtemperature region where the top and bottom voltages match they show that ρc vanishes
faster than ρab . To support the case where columnar defects enhance the c-axis correlation
Doyle et al also looked at the angular dependence of extracted ab and c-axis resistivities as
shown in figure 24. Both components are reduced for B// defects, as expected for strong
uniaxial pinning and finite-line tension. Apparent activation energies are obtained from
linear fit to Arrenhenius plots of the resistivities as a function of temperature. They found
that Ueff is field independent below the matching field, suggesting that vortices are indeed
ab
c
' Ueff
. If columnar defects enhance
localized on defects. Above the matching field Ueff
the line-like nature of BSCCO 2212 as suggested by uniaxial enhancement above 40 K
(as discussed in section 4.2.1) this appears to be inconsistent with the local electrodynamic
picture proposed by Doyle et al . The coincidence of the three-dimensional-like behaviour
reflected by uniaxial enhancement and two-dimensional-like scaling (Kes et al 1990), has
been discussed by Gray et al (1996, 1997) (see section 4.4.2). They concluded that vortex–
vortex interactions have not yet been taken into account properly and that the contradiction
can be reconciled by considering that the vortices are not strictly two-dimensional, but that
the CuO2 layers remain very weakly coupled, up to the highest temperatures.
4.3.2. Peak effects in I (T ). Kwok et al (1994) reported a peak in J (T ) below the sharp
resistive transition at Tm in twinned YBCO crystals, as illustrated in figure 25. Tang et
al (1996) reported similar behaviour when the field is aligned parallel to the ab plane.
This behaviour is interpreted as a softening of the shear modulus C66 which indicates a
precursor to melting (of Larkin domains, Larkin and Ovchinnikov (1979)) in the presence
of correlated disorder (Larkin et al 1995). The low-Tc -layered superconductor 2H-NbSe2
1626
L F Cohen and H J Jensen
Figure 23. The flux transformer data from two high-quality BSCCO 2212 single crystals. The
temperature dependence of the top and bottom voltages Vt and Vb respectively. After Safar et
al (1992a).
shows possibly related results for free flux-flow Hall effects, see Bhattacharya et al (1994).
Recently, Yaron et al (1996) reported that small-angle neutron scattering (SANS) from
the flux-line lattice in high-quality niobium crystals reveals drastic structural disordering
near the peak effect seen in the transport critical current. The flux-line lattice appears to
disorder as a function of applied field in a two-step process characterized first, by a complete
loss of long-range translational (hexatic glass) followed by a subsequent loss of orientational
order (vortex glass).
4.3.3. Vortex slush—melting as a function of drive current. The observation of multiple
regimes of behaviour caused by the interplay of pinning and current drive may relate to the
most recent discussions concerning plastic flow deep in the solid state. Worthington et al
(1992) first introduced the term vortex slush while investigating the effect of disorder. Three
YBCO crystals were examined. A ‘clean’ crystal with low screening current density which
was irradiated with 3 MeV protons with fluence of 1016 cm2 , a crystal which was heavily
twinned and point defected, with much higher current density, and finally a crystal which
Magnetic behaviour of superconductors
1627
Figure 24. Flux transformer data from a columnar-defected BSCCO crystal with a 0.5 T
matching field. The top V23 /I and bottom V67 /I resistances (see figure 21) are marked as points
on the graph. The calculated secondary apparent resistance from the top and c-axis voltages at
1 T, are marked as curves on the graph. The inset shows the temperature dependence of the
ratio of V67 /V23 and V37 /V23 at 0.5 T for the same crystal. Note that in this experiment the
numeration for the bottom voltage probes is reversed compared with that indicated in figure 21.
After Doyle et al (1996).
was irradiated with 1 GeV Au ions creating columnar defects. For these ‘intermediately
disordered’ YBCO crystals, two transformations (or a double shoulder) were observed in
the resistivity as a function of current, for small magnetic fields 0.1 T, as illustrated in
figure 26. The first being associated with the remnants of the first-order melting transition
Tm and the second, at lower temperatures associated with the transition to zero linear ρab
at Tg . At higher magnetic fields this double transition was washed out.
At temperatures between Tg and Tm , Worthington et al (1992) suggested an intermediate
vortex ‘slush’, which had finite-linear resistivity greatly reduced from the flux-flow liquid
state above Tm . The authors demonstrated that the upper transition Tm is dependent on
the current value or is a non-equilibrium ‘current-induced’ melting transformation. The
possibility of additional heating due to collision of vortices with pinning centres was
considered by Worthington et al (1992). They assumed that the disorder-induced heating
effect grows with increasing current drive. Note that this is in contrast to the Koshelev et
al (1996b) model which assumes that the effect of the fluctuating component of the pinning
force which produces ‘shaking’ of vortices and an associated ‘shaking’ temperature Tsh is
inversely proportional to the drive current and is only a well defined effect for drive currents
less than the critical current.
Later publications by Safar et al also referred to the vortex slush in YBCO as a region
where finite transverse correlation has set in as indicated by a sharp resistive jump, but longrange order has not been established throughout the system. Many crystals show a sharp
resistivity jump but ρab remains finite at lower temperatures. This is more likely to occur
in irradiated or twinned crystals and is probably not significant in very clean crystals where
the onset of long-range order coincides with the resistive jump as discussed in section 4.5.
The term vortex slush may describe the same type of inhomogeneous onset of flux flow
as studied by Bhattacharay and Higgins (1993) and Yaron et al (1995).
1628
L F Cohen and H J Jensen
Figure 25. (a) The Temperature dependence of the resistivity below Tm at different angles of
the magnetic field with respect to the c-axis and the twin planes. (b) Temperature dependence
of the critical current for different orientations of the magnetic field with respect to the twin
boundaries, showing the peak effect just below Tm . The peak is maximum when the field is
aligned to the twin planes. From Kwok et al (1994).
4.3.4. Dissipation in highly anisotropic systems. The question has been raised, whether
the origin of dissipation in the highly anisotropic systems is Lorentz force determined. Iye
et al (1989) and Woo et al (1989) reported that in BiSrCaCuO and TlBaCaCuO 2212 thin
films down to 15 K, the DC resistivity was independent of the angle between the magnetic
field and the transport current when both lay in the ab plane. The implication was that the
resistivity was independent of the Lorentz force. The angular dependence of the resistivity
could also be explained by the Kes model (refer to section 4.3.3). In contrast, Palstra et
al (1988, 1989) who were studying YBCO 123 single crystals, found a distinct resistance
anisotropy, implying that in the more three-dimensional systems the dissipation appears to
be Lorentz driven. Silva et al (1995) have shown that the high-frequency surface resistance
of YBCO behaves similarly.
Kadowaki et al (1994) investigated c-axis transport measurements with the field
orientation both parallel and perpendicular to the c-axis. It is found that the current
carrying ability in the c-axis is hindered most when the field is aligned parallel to the
Magnetic behaviour of superconductors
1629
Figure 26. (a) Typical resistivity versus current-density isotherms for a YBCO crystal after
3 MeV proton irradiation at a dose of 1016 cm−2 . (b) The linear resistivity extracted from the ρ
versus J isotherms in (a) at low current. (c) The melting current in MA m−2 versus temperature
defined as the current where δ ln ρ/δ ln(J ) is maximal. After Worthington et al (1992).
1630
L F Cohen and H J Jensen
Figure 26. (Continued)
c-axis and least, when it is aligned along the ab planes. Figure 27 illustrates the huge caxis resistivity broadening in BSCCO in finite field for the Lorentz force-free geometry. It
might be supposed that the magnetic field between the planes would decouple the planes and
ultimately destroy the Josephson coupling and that the force-free geometry would support
the larger current. This is the opposite of what is observed. The experimental observations
are understood to result from the fact that when the field is aligned along the c-axis the
phase fluctuations are more deleterious to the current. However, it is not completely clear
to what extend the G–L anisotropy masks the Lorentz force dependence.
The pronounced broadening of the resistive transition is a unique phenomenon. Although
there are several explanations based on the traditional idea of vortex motion (Gray and Kim
1993), several novel approaches have been introduced such as vortex–antivortex excitations,
thermal fluctuations of flux lines (Fastampa et al 1993) and superconducting fluctuations
(Tsuneto 1988, Ikeda et al 1991, Kadowaki et al 1994). See also Koshelev (1996a) for a
phase slip model for c-axis resistivity. On a cautionary note, the validity of the Lorentz
force–free resistive broadening for the H //c//I geometry will depend on the purity of the
current path. Any current deviating along ab planes will generate a force geometry.
4.4. The order of the melting transition
It is suggested that the order of the melting transition is a function of the static disorder in
the crystal. In clean samples with negligible pinning the vortex solid–melting transition is
expected to be a first-order transition between a vortex liquid and a well defined Abrikosov
lattice. It is thought that disorder drives the transition second order as assumed in the
vortex glass scenario (section 2.4.3). The influence of defects and their dimensionality on
the order of the transition and the nature of the glassy or solid phase are key questions still
undergoing clarification. If the disorder is correlated in one or more dimensions then it is
predicted that different kinds of glasses may occur such as a Bose or smectic glass (see
Blatter et al 1994b).
Magnetic behaviour of superconductors
1631
Figure 27. The logarithmic high-field c-axis resistivity behaviour of a BSCCO 2212 single
crystal for the Lorentz force-free geometry, B//I //c-axis. After Kadowaki et al (1994).
A second unresolved issue is whether the melting of vortex lattice occurs by means
of a single or two-stage process. A two-stage process could take the form of lattice
decomposition into a liquid of vortex lines, followed by a decoupling transition, when
thermal excitations destroy long-range correlation parallel to the c-axis. The order of the
events depends on field and temperature and the form of the decomposition and decoupling
lines. Jagla and Balserio (1997) discussed the circumstances under which c-axis correlation
or ab plane long-range order disappears, including how the anisotropy of the system affects
the order in which they are lost.
4.4.1. Evidence for a first-order transition. Evidence for a first-order thermodynamic
melting line, denoted Tm or Hm originates from coincidence of a sharp, hysteretic, resistive
transition, changes in latent heat, a jump or stepwise change in the reversible magnetization
M(B) and a frequency-independent peak in AC susceptablility.
The resistive transition. There are three features of interest in the resistive transition. The
sharp but hysteretic behaviour in low fields, the appearance of one or more shoulders (vortex
slush) and finally the broadening of these features in applied magnetic fields (Lorentz forcedriven dissipation).
As shown in figure 28, untwinned YBCO 123 crystals with very low disorder, were
reported to show sharp hysteretic resistive transitions by Safar et al (1992c, 1993), Kwok
et al (1992, 1994) and Charalambous et al (1993). The transformation line Tm (B) inferred
from these observations for H //c geometry coincides with the irreversibility line Tirr
1632
L F Cohen and H J Jensen
Figure 28. (a) Normalized linear resistance versus temperature for an untwinned YBCO crystal,
using a SQUID picovoltmeter. Note the hysteresis. (b) As in (a) but over a wide temperature
range. (c) Hysteresis width as a function of the field. After Safar et al (1992c, 1993).
extracted from oscillator experiments in similarly clean crystals. Tm (B) fits rather well to
some of the melting criteria models (Farrell 1994). There are various theoretical derivations
of the vortex-lattice melting line based on the Lindemann criterion, see for example Blatter
and Ivlev (1994) and Brandt (1989). Houghton et al (1989) predict a power law form
(which is the form followed by Hirr in fully oxygenated YBCO) that is best approximated
by
Bm (T ) = B0 (1 − T /Tc )α
where α 6 2.
(4.4)
Magnetic behaviour of superconductors
1633
Figure 28. (Continued)
Note that this equation has the same form as equation (4.2) which described the observed
temperature dependence of the irreversibility line in YBCO 123. This is also consistent with
the predictions of the three-dimensional XY model as discussed by Cooper et al (1997). In
clean YBCO crystals, it looks as though the Hirr line and the melting line coincide.
It is suggested that the resistive hystersis seen in figure 28(a), indicates superheating
and supercooling found at a first-order melting transition. Of course it is unclear whether a
non-thermodynamic quantity such as resistivity should follow the same hysteretic behaviour
as the internal energy. A compelling experimental paper (Jiang et al 1995) addressed this
point and concluded that resistive hysteresis is neither a sufficient nor necessary condition
for first-order melting. Part of the Jiang experimental results are shown in figure 29.
Superheating and supercooling would imply specific sub-loops which were not observed.
Waiting at point B0 or C0 indicated on the figure, the resistance was not seen to relax to a
new value which would be expected as a result of equilibrating the temperature. These and
other observations described in this paper provide counter evidence against the hysteresis
width being directly related to the latent heat.
Safar et al (1993) found a critical value of magnetic field in YBCO crystals, at which
the slope of the apparent phase boundary in the H –T plane changed. At fields greater
than the critical value, the sharp resistive transition was broadened and the hysteresis width
narrowed, as shown in figure 28(c). By examining the onset of nonlinear resistance Safar
et al also suggested that the glass transition Tg lies below the melting transition in crystals
which show the sharp resistive transition. Between these two phases the vortex system
had properties denoted ‘vortex slush’ where finite linear resistivity existed but was greatly
reduced from the flux-flow liquid state above Tm (Worthington 1992). The vortex slush is
discussed further in section 4.4.3. Various possible H –T phase diagrams were suggested
by Safar et al (1993) as shown in figure 30.
Remarkably, a sharp and hysteretic resistive drop in BSCCO crystals has been reported
1634
L F Cohen and H J Jensen
Figure 29. The history and time dependence of the resistivity hysteresis. The data points are
for partial heating and cooling cycles, the full curves are full heating and cooling data curves.
The inset is a schematic hysteresis and the corresponding subloops based on the assumption of
a first-order phase transition. After Jiang et al (1995).
Figure 30. A composite phase diagram for untwinned YBCO 123. The full circles are the
hysteretic melting temperatures Tm . Open squares are the vortex glass melting temperatures Tg .
Also shown are Hc2 and contours of constant resistance. The insets show three posssible phase
diagrams. After Safar et al (1993).
by Keener et al (1997) as shown in figure 31. Sharp features in ρ(T ) have also been
reported by Watauchi et al (1996) and Kadowaki (1996). The drop in BSCCO occurs
at a much lower resistance than in YBCO (0.02% of the normal state resistance versus
about 20% in YBCO) and is much harder to observe. Keener et al (1997) further claimed
Magnetic behaviour of superconductors
1635
Figure 31. The temperature dependence of Vab with I = 0.1 mA in magnetic fields of 0, 10,
20, 30, 40, 50, 70, 100, 120 Oe and higher labelled in the figure. The inset shows the electrode
configuration. After Keener et al (1996).
to observe a two-stage melting transition interpreting a second sharp feature in the liquid
phase as suggestive of thermal inter-layer decoupling of vortex lines. In the limit of low
current only, the decoupling and melting lines merge. Evidence of a sharp resistive drop in
BSCCO 2212 has also been discussed by Fuchs et al (1996), who found that the onset to the
resistive drop occurs concurrently with the equilibrium magnetization step/jump discussed
in the next section.
In the BSCCO 2212 system similar to YBCO 123, there is strong evidence suggestive
of a critical point, although it is unclear whether in fact data merely reflect a crossover in
pinning properties. The field at which it occurs in BSCCO varies from 30–2000 mT and
is sensitive to oxygen content (Ando et al 1995, Khaykovich et al 1996) which may be
tuning the intrinsic anisotropy (Kishio et al 1994) or the disorder. Pastoriza et al (1994a),
reported a frequency-independent Hirr line extracted from AC suceptability measurements,
which became frequency dependent above 36 mT. The reversible magnetization jump also
disappears at the same field (as discussed below). In agreement with this, Keener et al
(1997), found that the sharp hysteretic resistive drop broadens in fields greater than 70 mT.
The crossover field is independent of current, possibly reflecting thermodynamic behaviour.
However, the temperature Tm at which the sharp drop in resistance occurs, increases as the
current is decreased. This suggests that the resistance drop is associated with a currentdependent shearing mechanism rather than a sudden disappearance of C66 expected at a
melting transition.
More direct evidence for a loss of shear viscosity has been measured at Tm in a rather
novel experiment by Pastoriza and Kes (1995) where parallel tracks of columnar defects
were introduced in BSCCO crystals. In this configuration, the restoring shear force of
vortices situated in the weak-pinned channels between tracks, were measured by a simple
resistance measurement with the applied current perpendicular to the tracks. Within a
continuum approximation, the current density Js , which initiates the flow of vortices in
these channels can be expressed as Js = 2AC66 /W B where W is the width of the channel,
1636
L F Cohen and H J Jensen
A is a constant and B is the field. Below a certain field of the order of 30 mT, a finite shear
current density is identified in the weakly pinned material. The shear current for fields less
than 10 mT is indicated by an arrow in figure 32(b). No shear current is supposed to exist
for higher fields because it is assumed that the system is in the liquid state at higher fields.
Js is interpreted as the force needed to overcome the interaction of the vortex lattice with
the channel boundaries. For J greater than Js , the vortex system will comprise of pinned
vortices and flowing vortex channels. In fact this describes certain kinds of plastic motion
very well. In a sense, this is an observation of current-induced melting (see section 4.43).
Measurements of entropy change. For a sample exhibiting a first-order melting, there
should be a jump in the latent heat and in the magnetization 1M associated with a change
of entropy, such that
1M = [dTm /dH ]1S
(4.5)
where 1S is the entropy jump per unit volume. The latent heat L per unit volume and the
magnetization jump 1Mm are related to the entropy change by
L = H /sφ0 Tm 1Sm
(4.6)
1Mm = H /sφ0 (dTm /dH )1Sm
(4.7)
where s is the spacing of the CuO planes, Tm is the melting temperature and H is the
applied field in Oe and δSm is the entropy change per unit volume. For more extensive
discussions of this topic see Farrell et al (1995, 1996) and Rae (1996).
Magnetization. Experimental detection of the jump in magnetization is difficult because
the background magnetization change over the same temperature interval of the jump, is
about eight times greater than the magnetization jump itself. Nevertheless, a reversible
magnetization jump associated with an entropy change of 0.06kb , was first reported in
BSCCO crystals in 1994a by Pastoriza et al using DC SQUID magnetometry. Majer et al
(1995) repeated these experiments using local Hall sensor arrays. The data is extremely
clean, the discontinuous jump in magnetization is shown in figure 33. Also shown are the
estimated entropy change per vortex as a function of temperature. Close to Tc , δs appears
to increase rapidly, which the authors suggest could be attributed to critical fluctuations.
Farrell et al (1996) raised concern regarding the estimate of the entropy change in
the Zeldov paper, once the demagnetization factors have been taken into account. This has
been considered further by Rae (1996). If associated with a first-order transition, the Zeldov
experiments indicate that the density of vortices increases in the liquid state, resembling the
water–ice transition.
Pastoriza et al (1994a) carried out AC susceptibility using a SQUID as an amplifier.
The resulting low-field phase diagram showing a frequency-independent irreversibility line,
is shown in figure 34. As first discussed by Pastoriza et al (1994a) and confirmed by Zeldov
et al (1990) the transition line terminates at a critical point in low applied fields of the order
of 40 mT at 40 K. The phase diagrams for BSCCO 2212 that emerge are shown in figure 34.
This resembles the YBCO phase diagram (see for example figure 30) but shifted down to
lower temperature and field scales, as one might anticipate in the more anisotropic system
(Koshelev et al 1996).
Farrell et al (1995) repeated the Zeldov experiment using global SQUID magnetometry.
The magnitude of the δM change was found to correlate with the strength of the irreversible
signal as determined by varying the field orientation. This suggests that the change of
Magnetic behaviour of superconductors
1637
Figure 32. (a) The Arrhenius plot of voltage versus temperature at a current density of
106 A M−2 in a 10, 20, 30, 40 and 50 mT field (from left to right). Full symbols; before
irradiation, open symbols; after irradiation. The inset shows the zero-field transition. (b) I –V
characteristic for the irradiated samples at different magnetic fields in mT, at 80 K. The arrow
marks the estimated shear current. After Pastoriza and Kes (1995).
magnetization is due to the sudden disappearance of pinning, more likely related to a secondorder decoupling transition than a discontinuous entropy change. These results contradict
local AC suceptiblility measurements by Schmidt et al (1996) who claimed that the size of
1638
L F Cohen and H J Jensen
Figure 33. (a) A step in local B on crossing the melting line by decreasing the temperature at
a constant applied field of 50 Oe. The full curve is a guide to the eye. (b) Entropy change per
vortex pre-layer at the melting transition as a function of Tm . The inset gives an expanded view
near the critical point. The full curves show linear fit to the data. After Zeldov et al (1995a).
the jump in δM is independent of the angle of the magnetic field with respect to the c-axis
and also independent of frequency. This issue has yet to be resolved.
Zeldov et al (1995a) reported expressions (see references therein), for δs assuming the
transition at Tm to be either vortex lattice melting or decoupling. For melting, δs can be
calculated from the internal energy difference between a vortex solid and a vortex liquid
per unit volume, δU ' cL2 c66 where cL is the Lindemann number. Using an expression for
the melt temperature Tm ' 10.8cL2 c66 a03 where a0 is the intervortex spacing and = 1/γ
reflects the anisotropy, the entropy change per vortex per layer
δs ' (0.1d/)(Bm /φ0 )−1/2 .
(4.8)
Magnetic behaviour of superconductors
1639
Figure 34. The low-field phase diagram of BSCCO. From DC magnetization (full circles),
from the peak in the in-phase part of the differential susceptibility at selected frequencies. After
Pastoriza et al (1994a). (b) The first-order phase-transition line in BSCCO as measured by field
(circles) and temperature (squares) scans. The full curve is a fit to (1 − T /Tc )α vortex lattice
melting behaviour. The broken curve is a fit to (Tc − T )/T decoupling transition. The inset
shows the phase transition line Bm in the vicinity of the Tc . After Zeldov (1995a).
Hanaguri et al (1996) and Khaykovich et al (1996) explored the melting line as a
function of oxygen doping which reduces the anisotropy. Both found that the melting line
becomes steeper as the anisotropy γ is reduced, as illustrated in figure 35. Hanaguri et
al estimated δs from the size of the δM step. They found that by increasing the oxygen
content, the temperature δs increases but with increasing magnetic field, δs decreases. All
of these are contradictory to the simple vortex lattice melting picture and these results need
further explanation.
Welp et al (1996) reported clean results for the δM jump in untwinned YBCO crystals
using a global SQUID magnetometer. The change of M in fixed field (temperature) sweeping
the temperature (field) are consistent with the local slope of the melting line, in agreement
1640
L F Cohen and H J Jensen
Figure 35. (a) Magnetic phase diagrams of over-doped and optimally doped BSCCO 2212
single crystals. Open diamonds indicate the position of the second peak. The inset shows the
temperature dependence of δM. (b) The magnetic field dependence of δs. The curves are guides
to the eye. The inset shows the temperature dependence of δs. After Hanaguri et al (1996).
with equation (4.7). Figure 36 shows δM and δs versus temperature. Note that the rise in
δs close to Tc , reported in the BSCCO crystals is apparently absent in YBCO crystals.
As yet no existing theory can fully describe the temperature dependences or amplitudes
of Bm or sm in the YBCO and BSCCO system.
Magnetic behaviour of superconductors
1641
Figure 36. (a) The temperature dependence of the magnetization in 4.2 T. The dotted
curve represents a linear extrapolation of the low-temperature variation. The inset shows the
magnetization jump in 4.2 T and 2.9 T fields. (b) Top panel: the temperature dependence of
magnetization and entropy jump. Bottom panel: the phase diagram of the melting transition
from resistivity and magnetization measurements. After Welp et al (1996).
Specific heat. Specific heat directly measures a change in latent heat. Energy must be
supplied to the crystal as a whole, in order to drive the vortex assembly through the transition.
In YBCO this is more than two orders of magnitude greater than the expected latent heat.
Schilling (1996) looked at untwinned YBCO crystals using a differential thermal analysis
1642
L F Cohen and H J Jensen
Figure 37. (a) The temperature difference δ(Ts − Tr ) as a function of the sample temperature
Ts between the untwinned YBCO single crystal and a copper reference, measured in a magnetic
field of 5 T at various heating rates. The data is shifted vertically for clarity after corrections for
the smooth background differences in Cr − Cs which are qualitatively similar to the 80–85 K
segments displayed in the right inset. The right inset shows corresponding data taken in the
zero magnetic field around Tc . The left inset shows the experimental configuration with the heat
links ks and kr respectively. (b) The entropy change δS per vortex per layer. Full circles are
independent estimates from δM shifts at Hm on the same sample. The inset shows the first-order
phase boundary and the Hc2 (T ) crossover region which separates the normal and vortex fluid
states. After Schilling et al (1996).
technique (Schilling 1995) with a resolution in C better than 1 mJ/mole/K2 and in latent
heat L ≈ 40 µJ kB−1 . Their results are shown in figure 37. They reported an entropy change
δs ' 0.45 kB /vortex/layer, in agreement with the observed changes in 1M reported by
Welp et al (1996) on similar crystals. Based on the criteria set up at the beginning of this
section, these results appear to demonstrate rather clearly that a first-order phase transition
takes place in the vortex state of untwinned YBCO crystals.
However, Moore (1997), has drawn attention to the very different temperature-dependent
form for δs in the BSSCO and YBCO systems which is difficult to understand. Moore
proposed that the changes in magnetization, entropy and resistance discussed in this section,
may not be about a first-order transition but may reflect an underlying crossover from threedimensional to two-dimensional behaviour when the phase correlation length l along the
field direction in the vortex liquid becomes comparable with the sample dimension. Central
to the Moore picture is the idea that at all non-zero temperatures the system is in a liquid
state and therefore correlation lengths continue to grow at the crystal is cooled. Moore
Magnetic behaviour of superconductors
1643
Figure 37. (Continued)
calculated that l grows very rapidly as the temperature is lowered and the crossover in
dimensionality is quite sharp and comparable with the width of the drop in magnetization.
4.4.2. Decoupling transition. Various questions still need to be addressed concerning
whether the vortex lattice melts via a two-stage process. The influence of disorder (thermal,
current induced, static, etc) on this process and the role of electromagnetic rather than
Josephson coupling across the planes (Blatter et al 1996a, b, Nordborg et al 1996, Aegerter
et al 1996, Lee et al 1997).
Decoupling as a result of thermal disorder. Theoretically there are many predictions which
suggest that the vortex lattice melts via a two-stage process. First into a line liquid and then
at higher temperatures decoupling into a system where long-range correlation has been lost
along the c-axis. Well defined thermal decoupling of the planes associated with thermal
fluctuations of pancake vortices has been discussed theoretically Jensen and Minnhagen
(1991), Daemen et al (1993), Glazman and Koshelev (1991), Ikeda (1995), Li and Teitel
(1994) and Blatter et al (1994). It has been predicted that a change in c-axis transport or
C44 should occur at this thermally induced crossover. A crossover field is suggested B0
which takes a similar form in each theory.
B0 ≈ 4φ0 /γ 2 d 2
(4.9)
where d is the interplanar spacing and the temperature dependence of such a decoupling
line is
BD = B0 (Tc − T )/T .
(4.10)
Evidence for decoupling in the liquid state comes from ρc and flux transformer transport
data as discussed in section 4.4.1. There is strong evidence from combined ρc (T ) and AC
susceptibility (Pastoriza et al 1994a) or miniature two coil c-axis transmissivity (Doyle et
al 1995b) and mutual inductance techniques, (Ando et al 1995), which suggests that in
1644
L F Cohen and H J Jensen
pure unirradiated BSCCO crystals melting and decoupling occur simultaneously. At high
fields and low temperatures, Pastoriza’s AC susceptibility data show two dissipation peaks
(which are frequency dependent) in the transition to the reversible state. This was originally
interpreted as a two-step transition to an incoherent liquid phase, attributed to first a loss of
c-axis long-range correlation and then a depinning of pancakes at a field associated with the
DC irreversibility line, i.e. inter-plane and intra-plane dissipation occurring consecutively
(see Arribere et al 1993 and references therein). More recently these peaks have been
associated with the resistivity being different in the c-axis and the ab plane and the matching
of the skin depths (associated with the resistivity and measuring frequency) to some sample
dimension (Supple et al 1995, Steel and Greybeal 1992). Pastoriza et al (1994a) claimed
that the peak in the AC susceptibility measurement which occurs when the skin depth δ
matches a sample dimension D for a given frequency ω allowed the author to estimate the
c-axis resistivity at this transition, by using the following relationship
D = δ = ρ(T )c2 /2φ0 ω.
(4.11)
Pastoriza et al (1994a) proposed that the frequency-independent Hirr line below 36 mT is
a true first-order phase transition ending in a critical point at 36 mT (as shown in figure 34).
From the high electrical resistance along the c-axis they also claimed that this transition
coincides with a three-dimensional to two-dimensional crossover. Doyle et al (1995b),
explored this further by making miniature two-coil c-axis transmissivity measurements.
Figure 38 shows the imaginary part of the transmitted voltage. The real part measures ρc
and shows a sharp drop as a function of temperature or magnetic field. (Such sharp drops
in ρab are interpreted as melting, refer to figure 28(a).) Two loss peaks are also observed
in the imaginary transmitted voltage as a function of temperature and are attributed to ab
(the higher temperature peak) and c-axis currents, as before. As the magnetic field is varied
evidence of a sharp transition appears and moves through the c-axis current peak. This is
strongly suggestive of a transition which is associated with a sudden change in the local
c-axis resistivity, i.e. a decoupling transition.
In contrast to the above experiments, Wan et al (1994) flux transformer data taken
within a few Kelvin of Tc was suggestive of a two-step transition surviving up to Tc . It was
interpreted as in-plane dissociation by a Kosterlitz–Thouless-type process first and then a
Josephson decoupling transition. However, validity of these conclusions is doubtful because
data which was taken at very different effective electric field was compared directly. Cho et
al (1994), showed that c-axis transport measurements may also indicate a two-step transition
where the decoupling line lies above the melting line, in agreement with the Wan et al result.
Kadowaki et al (1994) investigated c-axis transport measurements with the field parallel
and perpendicular. From the very nonlinear and hysteretic I –C curves in applied magnetic
field, a field scale (B ∗ T ) is identified and represents a measure of the Josephson coupling
strength between adjacent pancake vortices. The authors claimed that at high temperatures
a short-range Josephson coupled–vortex liquid exists above the melting line. However,
a note of caution is required here because large transport currents were used in many of
the experiments which claim to observe a two-step transition and these currents may have
induced heating and other complications.
Gray et al (1996, 1997) addressed the influence of correlated disorder on the thermal
decoupling transition. In columnar-defected thallium 2212 thin films the films appear to
behave quasi-two-dimensional-like when the field is not closely aligned to the tracks. When
the field is aligned along or close to the tracks, directional suppression of the ab plane
resistivity is observed, implying vortex-line-like behaviour. The authors suggested that
there is always weak coupling between the plane even at temperatures above the thermal
Magnetic behaviour of superconductors
1645
Figure 38. The imaginary transmitted voltage at 10 kHz with DC fields of (a) 70 mT, (b)
60 mT, (c) 40 mT and (d ) 10 mT applied parallel to the c-axis. The peak at higher temperatures
is associated with in-plane resistivity and that at lower temperatures with out-of-plane currents.
After Doyle et al (1995b).
decoupling temperature and they also implied that vortex–vortex interactions have not been
properly considered in the case of columnar-defected materials.
Decoupling in the presence of current-induced disorder. Many experiments which might be
considered to be in a high-current limit, are suggestive of a two-step transition process. See
for example, Wan et al (1993) and Cho et al (1994). Keener et al (1996) showed evidence
that the two-step process may be current induced. The statement concerning experiments
conduced at high current, discussed in the last section is also valid here.
1646
L F Cohen and H J Jensen
Decoupling in the presence of static disorder. There have been many experiments discussed
throughout the text which could be described in this section. Here we focus on the
debate concerning the fate of the melting/decoupling line as the crystal is cooled such
that underlying static disorder pin vortices and destroy the perfect Abrikosov lattice. Note
that magnetically induced decoupling is not a dimensional crossover as such but will occur
in anisotropic systems if the pinning in each layer is sufficiently strong. In this case, the
correlation between the pancake vortices in the c-direction is destroyed above a crossover
field B2D , because the magnetic repulsion between pancakes in the same layer becomes
stronger than the attraction in adjacent layers.
From various experiments described in sections 4.6.3 and 4.7 it is established that in
unirradiated BSCCO, bulk pinning becomes effective at around 40 K and at a crossover
field H ∗ (see for example Zeldov et al 1995b, Cohen et al 1997). Khaykovich et al (1996)
found that below about 40 K the transition to bulk pinning occurs very sharply at the local
field B ∗ which is approximately temperature independent and terminates at the critical point
(discussed in section 4.5.1). Neutron diffraction data from Cubitt et al (1993) suggests that
an ordered vortex lattice (recently denoted Bragg glass), exists in the entire low-field phase
below H ∗ . This has led Khaykovich et al to suggest that although Bm is a simultaneous
first-order melting and decoupling transition, in the presence of pinning, this becomes a
sharp second-order decoupling transition at B ∗ . This area has to be explored further.
Josephson weak-link behaviour. Passing a transport current along the c-axis would appear
to be the most straightforward method to examine the dimensionality issue. From caxis transport measurements Kleiner et al (1994a, b) have shown that Josephson weaklink characteristics and Shapiro steps can be obtained in every system at 4.2 K except
well oxygenated YBCO 123 crystals. Measurements have not been made at low magnetic
fields because of severe heating effects. It is quite probable that Josephson behaviour
can only occur once the vortices are pinned, because depinned vortices are susceptible to
flux flow which must introduce large phase fluctuations between planes. It is difficult to
draw firm conclusions when Josephson behaviour is not observed because of the number of
possible explanations. An absence of characteristic Josephson behaviour as a function of
temperature could imply a complete loss of c-axis correlation, an onset of very strong caxis correlation, depinning or melting into a highly entangled flux liquid (where flux cutting
and reforming must also introduce large plane–plane phase fluctuations). The temperature at
which Josephson behaviour disappears depends on the family of crystals and the angle of the
external magnetic field relative to the crystal planes. However, the absence of Josephson-like
features in well oxygenated YBCO crystals implies that quasi-two-dimensional behaviour,
or decoupling, does not occur in the irreversible regime. No exploration of the effect of
twinning on the observation of weak link behaviour in YBCO crystals, has been reported.
The c-axis conductivity in BSCCO 2212 crystals shows that there is a well defined
Josephson current at the field of the order of 2T for H //c for a temperature range between 20
and 10 K but that this current shows an anomalous re-entrant behaviour at lower temperatures
and a gradual smoothing out of the Josephson characteristic (Rodriguez et al 1993, Doyle et
al 1995a). The re-entrant behaviour has a dependence on magnetic prehistory and might be
attributed to an interference between shielding and transport currents (Gordeev et al 1994).
4.4.3. Critical scaling. The general description of critical lengths and timescales at
a continuous phase transition were set out in section 3.5, as were the current–voltage
characteristic (IVC) predictions. Evidence for glassy behaviour can perhaps be better
Magnetic behaviour of superconductors
1647
Table 1. Critical exponents.
v(z + 2 − d)
from ρL (T )
ξvg =
(ckT /ϕ0 J )1/2
1.7
6.6
1.6
6.4
1.8
6.4
1.8 ± 0.2
6.2
6.5
6.5
6±2
0.3 µm
2±1
6.5
15 µm
z from I –V v from
curve
J0+ (T )
v(z + 2 − d)
from z, v
Sample
Comments
Field T
YBCO
Film (1989)
Koch
T ↓ Tg
2//c
3//c
4//c
1//c
4.9
5.0
4.7
1–600 MHz
T ↓ Tg
1–3//c
6//c
3.7 ± 0.46
3.4 ± 1.5
T ↓ Tg
> 10//c
ρ = (T − T ∗ )S
s =6±1
T ↓ Tg
6//ab
s = 1.35 ± 0.15
e-irradiated
T ↓ Tg
1–8//c
s = 1.2–3.0
depends on field
and is not
glassy-like
Film Dekker (1992) T ↑ T g
Film Hui Wu (1993)
Twinned
crystal
Gammel (1991)
Untwinned
crystal
Safar et al (1993)
Untwinned
YBCO crystal
Kwok et al (1994)
Untwinned YBCO
crystal
Fendrich (1995)
BSCCO
BSCCO crystal
Doyle et al (1996)
BSCCO crystal
Zech et al (1995)
BSCCO films
Miu et al (1995)
BSCCO crystal
Safar et al (1992b)
4.3 ± 1.5
Below 20 K
T ↓ Tg
columnar defected
T ↓ Tg
columnar defected
T ↓ Tg
T ↓ Tg
c-axis transport
µ depends
on field
smoothly
s = v(z − 1)
s = 8.5
ab-plane transport 8.5
ab-plane transport 9
2–6//c
6.7
examined well below this transition line. We discuss the evidence for glassy behaviour deep
inside the solid, in section 4.6. It is appropriate here to mention that there are problems
associated with the scaling experiments which have been long understood from computer
simulations of continuous transitions. If the experimental window is too small conclusions
might well be misleading.
If critical exponents can be extracted reliably then this suggests that there is a
thermodynamic transition and it is second order. The critical exponents must be independent
of magnetic field. Critical exponents which appear to vary smoothly with magnetic field are
more suggestive of plastic behaviour. From exponents determined from below the transition,
such as Dekker et al (1992), or in electron-irradiated crystals, there is definitely evidence
of plastic behaviour. In table 1 we examine the consistency of extracted exponents between
different samples. There is some agreement between YBCO films, twinned and untwinned
YBCO crystals (at low temperatures and high fields in the latter case). The exponents change
systematically for different kinds of static disorder, such as columnar defects or intrinsic
planes. Note that in the case of intrinsic planes theoretical studies of inter-layer vortex
melting in two-dimensional layered systems predict that an intrinsically pinned vortex lattice
cannot melt via a second-order transition (Mikheev and Komomeisky 1991, Korshunov and
Larkin 1992). Usually IVCs can only be scaled in a range of temperatures about 2 K wide
around the transition for YBCO and about 5 K wide for BSCCO. Occasionally scaling of
1648
L F Cohen and H J Jensen
transport IVCs are reported to occur over a wider temperature range which counts against
the existence of a critical region (Koch et al 1989). Scaling also occurs over a narrow range
of currents only. For small currents, it appears that the equilibrium properties are lost and
the vortices become pinned. Large currents have been shown to induce non-equilibrium
effects when measuring the resistivity (Liang et al 1996).
Four representative experiments are examined in more detail below.
YBCO films–H //c. Scaling was first shown to occur in YBCO thin films (Koch et al 1989)
although the strength of the scaling argument was criticized by Coppersmith et al (1990)
and Griessen (1990). In order to be in the critical regime, restrictions on length-scales were
first that ξ > l, where l is the average distance between vortex lines, which is only satisfied
in high fields (but below Hc2 ), and secondly that ξ 6 t, where t is the film thickness,
otherwise the three-dimensional assumption would also break down. Koch claimed that the
I –V curve shape for T 6 T g was inconsistent with the standard flux creep model, which
predicts that V ∼ sinh(I /I0 ) resulting in an IVC with a positive curvature on a log I –log V
plot. Figure 39 shows that the curvature is negative at low temperatures, with a value of
µ ∼ 0.4 ± 0.2, where µ is defined in equation (3.20).
Untwinned YBCO crystals H //ab. Kwok et al (1993, 1994) showed that the sharp resistive
transition associated with first-order melting is suppressed when the magnetic field is aligned
within 0.5◦ of the ab planes. For the H //ab geometry critical exponents are extracted by
plotting [d(ln ρ)/dT ]−1 versus T , where the slope of the straight line is 1/ν(z − 1) and the
intercept defines the transition temperature Tg as shown in figure 40. The exponents are
consistent with a smectic transformation as found in liquid crystals.
Unirradiated BSCCO crystals H //c. The IVCs of clean unirradiated BSCCO crystals are
expected to be linear above the decoupling field H ∗ (see sections 4.5.2 and 4.6.3). By
examining the temperature dependence of the linear resistivity Tg can be predicted from
equation (3.17). Safar et al (1992b) observed Arrhenius behaviour (i.e. ln ρ ∝ 1/T ) at
high temperatures and a critical scaling regime at low temperatures. Figure 41 shows the
mechanism by which the Tg line was extracted for pure BSCCO crystals. This is the same
as just described for the unirradiated YBCO crystals. Safar et al defined a temperature T ∗
below which the behaviour entered the critical regime. The critical exponent agrees with
vortex glass prediction.
Columnar-defect BSCCO crystals H //c. For correlated disorder such as columnar defects,
the low-temperature vortex solid is proposed to be a Bose glass (Nelson and Vinokur 1992).
Critical scaling analysis has been applied to the in-plane resistivity at high temperatures
by Miu et al (1995) and the c-axis resistivity by Seow et al (1996). The results from
the two experiments are consistent and produce critical exponents which agree with Bose
glass scaling. Seow et al found that for B parallel to the tracks, the linear resistivity along
the c-axis, ρc , does not show Arrenhius-like behaviour, whereas when B is misaligned
with the tracks, the resistivity becomes Arrenhius-like below about 1% of the normal state
resistivity. The behaviour of ρc , when the field is parallel to the tracks returns to Arrenhiuslike behaviour for B > Bφ . As shown in figure 42, when the field is aligned parallel to the
tracks and B < Bφ , ρc can be replotted in the critical scaling form, producing a value for
Tg = 66 K which is greatly shifted up in temperature compared with the virgin crystals.
The effective exponent s = ν 0 (z 0 − 2) = 8.5, agrees with Bose glass predictions.
Magnetic behaviour of superconductors
1649
Figure 39. The I –V curves at constant T for (a) H = 0.5 T and (b) H = 4 T. The curves
differ by intervals of 0.1 and 0.3 K, respectively. After Koch et al (1989).
4.5. Below the irreversibility line—the vortex solid
The conclusion drawn from the transport measurements (reviewed in section 4.4.3) are that
in the presence of disorder IVCs become nonlinear in a way that suggests transitions into
glassy-like solids. This is true for both YBCO and BSCCO. Exotic solids such as smectic
glasses, Bose glasses and Bragg glasses reflecting the dominant source of static disorder
were identified. The focus of this section is to examine evidence for the continuity of
behaviour below the irreversibility line. Vortex behaviour deep inside the vortex solid can
only be explored with magnetic measurements, where low electric fields can be accessed
1650
L F Cohen and H J Jensen
Figure 40. (a) The plot of 1/[(1/ρab )(dρab /dT )] versus T , for a YBCO crystal in a 6 T field.
The kink associated with the first-order melting is present for small misorientation angles of
0.5◦ , but disappearing when H is aligned to the planes. (b) The plot of H //ab versus T ∗ . The
inset shows the field dependence of the dynamic scaling exponent s obtained from (a). After
Kwok (1993).
easily. For reviews on magnetic relaxation see Yeshurun et al (1996), for thermally activated
motion see Schnack (1995), and for quantum creep see van Dalen (1995). As discussed in
section 3, the aim of monitoring dynamic behaviour is to determine the functional form of
the effective pinning barrier on current Ueff (J ).
4.5.1. Peak effects in J (B). Peaks in J (T ) were discussed in section 4.4.2 and they are not
associated with the phenomena discussed in this section. Kobayashi et al (1995a, b) showed
that crystals which show large J (B) peaks with a straightforward monotonic temperature
dependence, do not show the J (T ). Crystals with much weaker pinning with a nonmonotonic temperature dependence, show both a J (T ) peak and a J (B) peak, but they lie
in different positions in the H –T plane. The two peaks are not related but may coexist.
Magnetic behaviour of superconductors
1651
Figure 41. (a) The plot of the inverse logarithmic derivative of the resistance. The full curve
represents a fit to the vortex glass theory with Tg = 20.2 K. Deviations from the vortex glass
theory are observed above 28 K. The inset shows the critical exponents for different fields.
(b) The H –T -plane showing the positions of Tg and T ∗ lines. After Safar et al (1992b).
Peak effects were first discussed by Le Blanc and Little (1960) who observed an
anomalous peak in J (T , B) in LTS. Pippard (1969) and Larkin and Ochinnikov proposed
that this peak effect was related to softening the vortex lattice. (A soft lattice can pin more
strongly that a more rigid one and hence can produce current enhancement.) In fact there
are many mechanisms which can generate a peak in J , see Cambell and Evetts (1972) for
a summary.
Most YBCO crystals show an anomalous second peak in the magnetization loop, known
as the fishtail peak, illustrated in figure 43. The peak position has a strong temperature
dependence and in deoxygenated YBCO 123 and in YBCO 124 crystals it exists down
1652
L F Cohen and H J Jensen
Figure 42. The plot of ρc (dρc /dT )−1 and normalized ρc against temperature in a 0.7 T applied
field. The linear regime is clearly seen below 75 K. The inset shows the exponent n of the Bose
glass phase extracted from the resistivity data. After Seow et al (1996).
Figure 43. The M–H -loop in an untwinned YBCO crystal at 77 K, showing the second peak
feature denoted the fishtail peak. The field at which the peak maximum occurs has a strong
temperature dependence (1 − T /Tc )3/2 .
to the lowest measured temperatures. The origin of this feature has been much discussed
and there is still a lack of consensus in the literature. Explanations include the effects
of macroscopic granularity and underlying defect structure thought to be associated with
oxygen vacancies (Daümling et al 1990, Yeshurun et al 1994b, Osofsky et al 1992, Erb
et al 1996); simple dynamic effects associated with creep (Cohen et al 1993, Delin et al
1992, van Dalen 1995); Krusin-Elbaum et al (1992) discussed the J (B) peak in terms of a
a crossover from single vortex pinning to a pinning of vortex bundles; Perkins et al (1995)
suggested that it is related to the interplay between the field dependence of the characteristic
Magnetic behaviour of superconductors
1653
Figure 44. The M–H -loop in a BSCCO crystal at 30 K, showing the second peak feature
denoted the arrowhead peak. The peak only occurs over a limited range of temperatures and is
approximately independent of the field, as indicated by the line labelled Bsp in figure 51.
energy and current scales; Zhukov et al (1995) speculated that it is associated with plasticity
or with softening C66 ; and Abulafia et al (1996) implied that it is related to a crossover from
elastic to plastic vortex behaviour. Very pure untwinned YBCO crystals do not show this
feature and also highly disordered thin films do not show it. A rough measure of the local
static disorder in YBCO crystals is the value of the screening current density J , at 77 K and
1 T. Typical values for untwinned, twinned and proton irradiated YBCO are J = 102 , 103
and 104 A cm−2 , respectively. Werner et al (1994) reviewed the effect in many different
samples and concluded that it is caused by an interaction between the flux-line lattice and
the defect structure and may not be related to a specific defect structure itself. Erb et al
(1996) showed how the peak could be reversibly induced by introducing oxygen vacancies
in untwinned YBCO crystals. The effect of point defects and twin planes on the shape of
the fishtail feature was elucidated by Küpfer et al (1996).
In BSCCO 2212 crystals, an anomalous second peak is observed in J (B), and because
of its shape it is known as the arrowhead feature, as shown in figure 44. Unlike the fishtail
peak found in YBCO crystals, the arrowhead feature occurs between approximately 20 K
and 50 K only and the field at which it occurs is almost temperature independent. It can
be altered in size and position by increasing the number of point defects through electron
irradiation (Chikumoto et al 1992), by high pressure (Yang et al 1994) or low-temperature
oxygen annealing (Kishio et al 1994), by partial doping of lead onto the barium sites (Cai
et al 1994), and by introducing structural defects (Yang et al 1993b). As discussed in
sections 4.5.2 and 4.6.3, the field at which the arrowhead peak occurs, is associated with
magnetic decoupling of the vortex lattice. Zeldov et al (1994) showed that there is a strong
interplay between surface and bulk pinning effects at the peak field. It was first suggested by
Chikumoto et al (1992), and later by Yeshurun et al (1994a), Cai et al (1994) and Cohen et
al (1997), that the arrowhead feature results from an interplay between static and dynamic
effects. Kishio suggested that crystals which do not show the arrowhead feature (and there
are many which do not), may be so anisotropic that the peak field is unmeasurably low.
1654
L F Cohen and H J Jensen
Alternatively they may have an inhomogeneous distribution of properties resulting in a very
gradual change rather than a sharp crossover in behaviour as a function of field.
4.5.2. Unirradiated YBCO 123. From transport measurements we learn that above the
Hirr line, critical scaling suggests that second-order transitions take place in various elastic
vortex solids. The vortex glass exponent µ (defined in equation (3.20)) has been measured
in YBCO thin films by Dekker et al (1992) and Berghuis et al (1996), from below the Hirr
line. In both papers it was reported that the µ value, rather than change abruptly it slowly
varied between 0.19 and 0.94 as a function of temperature, magnetic field or current. The
Dekker results are shown in figure 45. Both the Dekker and the Berghuis observations are
important because they imply that the pinned vortex system does not necessarily appear
glassy, close to the Hirr line when determined from below it. That the so-called glass
exponent varies continuously with field implies some kind of plastic behaviour.
Turning to magnetic measurements, the first general discussion of evidence of glassy
behaviour in YBCO, came from Malozemoff and Fisher (1990). They drew attention to a
temperature-independent plateau in the normalized creep rate S(T ). As shown in figure 46,
the plateau appeared to have a universal value of 0.03 at fixed fields of the order of 1 T.
Expressing the normalized creep rate S = 1/[µ ln(t/t0 )] and substituting µ = 1, at an
attempt time of the order of 10−10 s the authors obtained a value of S = 0.033. The value
for µ and the attempt time are consistent with vortex glass and collective pinning theories.
Malozemoff (1991) also attributed the linear behaviour of S(T ) at temperatures below the
plateau to Anderson–Kim-type thermal activation and above it, to a softening of the glass,
possibly suggestive of plastic behaviour in agreement with the transport measurements
described above. Note that Caplin et al (1995), gave a useful explanation of why S = 0.03
could be so frequently observed simply as a result of the similarity in experimental conditions
in which the measurements are made (such as sample size electric and magnetic field ranges
etc). In the collective pinning model, many regimes of behaviour are possible in the vortex
solid (refer to Blatter et al (1994b)). Krusin-Elbaum et al (1992) presented evidence for
many of these regions from flux creep data in twinned YBCO crystals.
The same regimes of behaviour set out by Malozemoff have been further explored as a
function of magnetic field and temperature in twinned and untwinned 123 and 124 crystals
(see Cohen et al 1994b, Perkins et al 1995, Zhukov et al 1995). The Malozemoff ‘plateau’
in S(T ) observed at fixed field actually occurs over a wide range of temperatures and fields
as shown in figure 47, as region 1. Region 1 indicates where a glassy-like solid exists
in the H –T plane of well oxygenated YBCO. This region ‘shrinks’ as the crystals are
deoxygenated and made more anisotropic as discussed by Cohen et al (1994c). Perkins
et al (1996) discussed that by using equation (3.12), the dynamic normalized creep rate
Q = (d ln J /d ln E)B,T can be expressed as
d ln Ueff
(4.12)
S = 1/C
d ln J B,T
where C = ln(Bωd/E). A power law Ueff (J ) ∼ J −µ with µ independent of B and
T automatically results in constant S(B, T ). This implies a convex ln E–ln J curve, as
observed in transport measurement. The form of the ln E–ln J curve in this regime has
been confirmed over a large electric field window by a mixture of relaxation and transport
measurements, by Gordeev et al (1994), as illustrated in figure 48. Interestingly the authors
confirmed that the fishtail feature survives in transport measurements at high electric fields.
The region below the Malozemoff plateau in S(T ), is often analysed unreliably because
it only occurs at low applied field and self-field effects dominate. The region above the
Magnetic behaviour of superconductors
1655
Figure 45. (a) The temperature and (b) the field dependence of the critical exponent µ, on a
variety of thin-film YBCO samples. The temperature scale is normalized to the glass temperature
in each case. After Dekker et al (1992).
plateau in S(T ) was identified by Malozemoff as a softening of the glass. This region is
marked on the H –T plane in figure 47 as region 2. It has also been shown by Cohen et al
(1994b) that S(B) is linear in this region. Using the magnetic scaling analysis Perkins et
al (1995) showed that the linear S(B) is related to a logarithmic law Ueff (J ) dependence.
Logarithmic Ueff (J ) implies a power law IVC of the form E = J n , where in this case n
is inversely dependent on B and T . The behaviour of this regime has been compared with
various theoretical predictions from collective pinning as shown in figure 49. It is found
that the field dependence of U0 and J0 are not in agreement with that theory in its present
form. Abulafia et al (1996) interpreted the behaviour, in terms of plastic creep resulting
from dislocation flow. This is consistent with the Dekker et al (1992) and Berghuis et al
(1996) transport results discussed at the beginning of this section.
To summarize, except for regime 1, which survives up to high temperatures at low
1656
L F Cohen and H J Jensen
Figure 46. Normalized relaxation S versus T for a variety of YBCO samples at 0 T, 1 T and
2 T fields, illustrating the universality of S = 0.03. After Malozemoff et al (1990).
fields (as indicated in figure 47), most dynamic magnetization techniques probably measure
plastic behaviour associated with the the static disorder and inhomogeneity of a particular
crystal. In general it is the field dependence of the creep rate S(B) or Q(B) which fails
to fit into the framework of collective pinning. This is a reflection of the fact that most
magnetization measurements are set up such that the interplay between vortex–vortex and
vortex pin energy is conductive to plastic flow over most of the H –T plane in YBCO. This
is discussed in detail by Zhukov et al (1995) and Abulafia et al (1996).
4.5.3. Unirradiated BSCCO 2212. Using transport techniques to carry out critical scaling
analysis, Safar et al (1992b) suggested that glassy behaviour occurred below 20 K. However,
from the magnetization measurements there is no evidence for the Malezemoff plateau in
S(T ) or Q(T ) in BSCCO at these temperatures. It is generally agreed from the symmetry
of the M–H loop shape and the size of the irreversible signal that below 20 K, bulk
pinning dominates over surface or geometric barrier effects. Magnetic scaling analysis can
be performed below 20 K in BSCCO 2212 and the results implying power law E–J curves
as found in region 2 in YBCO. Totty et al (1996) found that m and n, the field dependence
of the characteristic current and energy scales, are also similar to YBCO but with a stronger
temperature dependence. Given the similarity to YBCO, the observed behaviour in BSCCO
Magnetic behaviour of superconductors
1657
Figure 47. An experimentally derived H –T diagram showing regimes of flux creep behaviour.
Hp is the critical state penetration field, below which the sample is not fully penetrated, Hd is
the field at which S begins to rise linearly off the 0.03 plateau, Hs is the field at which much
faster creep occurs and Hirr is the irreversibility line. After Cohen et al (1994b).
Figure 48. Current and voltage characterisitics of a YBCO crystal at 87 K at several magnetic
fields. The higher electric field data is compiled from direct electric transport and the lower
electric field data is from magnetization measurements. After Gordeev et al (1994).
2212 over most of the H –T plane is also probably some kind of plastic response.
van Dalen et al (1996) explored the dynamic creep rate Q variation as a function of an
angle in unirradiated BSCCO crystals at 20 K and found that the measured current density,
Q and Uc , the characteristic pinning energy, scale with the c-axis component of the external
field. Nideröst et al (1996), observed three regimes of flux creep behaviour measured by
long-time relaxation over seven decades of time, as a function of temperature. Using the
1658
L F Cohen and H J Jensen
Figure 49. (a) The predicted values of the exponents m and n for the power-law field dependence
of J0 ∝ B m and U0 ∝ B n corresponding to each of the regimes in figure 3, where sb, lb and
CDW denote small bundle, large bundle and charge density wave. See Blatter et al (1994) for
definitions of these regimes. The data for a twinned Tm 123 crystal is indicated by the arrow.
After Perkins et al (1996).
Maley method they found a logarithmic Ueff (J ) function below 20 K and a power law
Ueff (J ) function above 40 K. Between 20 K and 40 K no unique functional dependence
could be found. The low temperature behaviour is attributed to individual two-dimensional
pancake vortex pinning. Several papers from van der Beek et al (1992), and Vinokur et
Magnetic behaviour of superconductors
1659
Figure 50. The B–T phase diagram where the full curve is the theoretical three-dimensional
melting line (Houghton et al 1990), the circles are the melting transition from the neutron
diffraction intensity and the squares show the boundary between the reversible and irreversible
magnetic behaviour in hysteresis loops. After Cubitt et al (1993).
al (1995), have also addressed the dynamics in BSCCO and concluded that dislocation
mediated creep rather than two-dimensional collective pinning provides a good description
of the magnetic relaxation. So in this respect there is some consensus about the behaviour
of BSCCO below the irreversibility line.
Other information has been obtained about the form of the H –T diagram in BSCCO
2212. The vortex lines can be regarded as two-dimensional pancake vortices confined to the
Cu2 O layers by Josephson and/or magnetic coupling. At low fields, such coupling results
in essentially three-dimensional flux lines. Josephson coupling ensures phase locking or
phase coherence between pancakes on adjacent layers. At high fields the in-plane repulsion
between pancakes exceeds their inter-plane attraction. Uncorrelated pinning in different
Cu2 O layers breaks up the flux lines in the field direction, leading to so-called flux-line
decomposition, or decoupling. Such decomposition has been inferred from small angle
neutron diffraction experiments by Cubitt et al (1993) and µSR experiments by Lee et
al (1993, 1997). The signature for a correlated three-dimensional lattice disappears above
60 mT. Vinokur et al (1990) predicted this decomposition to occur at
B2D = φ0 /(sγ )2
(4.13)
where s is the spacing and γ is the anisotropy factor. The field associated with the
decomposition line coincides with the arrowhead peak.
1660
L F Cohen and H J Jensen
Figure 51. The phase diagram of BSCCO showing the penetration field Hp , the onset of
irreversible shielding HIS , the bulk irreversibility line BIR and the low-field phase transition at
the onset of the second peak (arrowhead peak) Bsp . The HIS line is associated with surface and
geometric barriers and the fits to theoretical forms for these lines are shown. Refer to Zeldov
et al (1995b).
Rodriguez et al (1993) reported a gradual smoothing out of the Josephson characteristic
below 10 K. From these experiments as well as AC susceptibility and DC magnetometry, de
la Cruz et al (1994a), suggested that the three-dimensional solid exists up to high magnetic
fields at the lowest temperatures, modifying the original Cubitt et al picture. In fact more
recent neutron and muon data also support this claim Aegerter (1996), Bernhard et al (1995).
Zeldov et al (1995b) produced a detailed phase diagram based on local magnetization
measurements using Hall bar arrays. The high-temperature melting line was discussed in
section 4.4.1. At temperatures below the suggested critical point, bulk pinning starts to be
important and the non-equilibrium phase diagram is extremely sensitive to the dimensionality
of the pinning and the superconducting anisotropy.
4.5.4. Irradiated YBCO 123 and BSCCO 2212 crystals. Irradiation enhances the screening
current density, alters the position of the fishtail or arrowhead peak and has been seen
to enhance or suppress the position of Hirr line in the H –T plane. YBCO crystals show
unique lock in signatures to twin planes (Oussena et al 1996, Zhukov et al 1996), to CuO2
(intrinsic) planes and to columnar defects. As a function of field orientation the Hirr line in
both YBCO crystals (Krusin-Elbaum et al 1994a) and BSCCO crystals (Zech et al 1995)
display characteristics which resemble the predicted Bose glass cusp at high temperatures.
Klein et al (1993a) discussed ‘flux flop’ effects associated with locking onto columns at
low fields and small angles away from the c-axis, have been also been reported. Hardy et al
(1996), studied the accommodation of vortices to tilted line defects with various electronic
anisotropies from crystals of 2212, 2223, 1223 and 123 composition and also present a
brief review of the subject. For both YBCO and BSCCO at low temperatures isotropic
pinning enhancement is observed. Directional effect are observed at higher temperatures.
The isotropic regime is ascribed to vortices zig-zagging between the ab planes and the
Magnetic behaviour of superconductors
1661
Figure 52. The persistent current density J and normalized relaxation S as a function of
temperature, for two different irradiated YBCO crystals where α is the angle of the columns
with respect to the c-axis and Bφ is the matching field. After Civale et al (1996).
columns, keeping their mean-field direction along the applied field. The model invoked by
Hardy can explain the data but assumes that the vortices are line-like and have line tension
at all temperatures for all crystals studied. This is then in conflict with the concept that
BSCCO is two-dimensional-like at low temperatures. There are few published systematic
studies of the dynamics of vortices in columnar-defected YBCO or BSCCO crystals.
Irradiated YBCO 123. There are many papers on the effect of irradiation on flux dynamics.
Initially the influence of point-defect irradiation was studied in YBCO for example by Civale
et al (1990) using 3 MeV protons, and it was found that although the critical current was
enhanced, the irreversibility line and creep rates (and therefore pinning potential) were
almost unaffected. Thompson et al (1991a, 1993) took the proton irradiation YBCO studies
further, using the Maley analysis to extract a functional form for the Ueff (J ) function which
1662
L F Cohen and H J Jensen
agreed with collective pinning theory. These experiments were only carried out at one
fixed field of 1 T. They concluded that the quasi-exponential temperature dependence of
the current density results from flux creep and is not inconsistent with collective pinning
theory. Sun et al (1992), examined both the temperature and field dependence of the
activation energy in proton irradiated YBCO. Ueff (H ) was found to vary as H −α and α
depended on both temperature and current.
Schindler (1991) found that fast neutrons with energy greater than 0.1 MeV increased
the critical current and decreased the creep rate, implying a change of the pinning potential
in YBCO crystals. Konczykowski et al (1991) found that irradiating with 5.3 GeV lead ions
increased the Hirr line dramatically and the critical current and also decreased the creep rate.
Civale et al (1991a, b) found similar current and Hirr line enhancement from discontinuous
tracks of amorphous material produced by 580 MeV Sn ions at 30◦ to the c-axis. They
called these tracks columnar defects. The effectiveness of the tracks were explored as a
function of angle of applied field, irradiation dosage and temperature. It was found that the
tracks were most effective when the field was aligned parallel to the tracks. Above 88 K,
Hirr was independent of dose and similar to the unirradiated crystal.
Flux dynamics of columnar-defected twinned YBCO crystals was first studied by
Konczykowski et al (1991, 1993). Long-time relaxation was found to be non-logarithmic
exhibiting an increase in effective barrier for flux creep with decreasing current in agreement
with vortex loop nucleation as proposed by Nelson and Vinokur (1992). The experiments
were only made at very low applied fields of the order of 50 mT. Long-time relaxation
measurements have since been made by Civale et al (1996) at fields less than the matching
field and by Thompson et al (1997) at fields both less than and greater than the matching
field. Both groups report that for fields less than the matching field the normalized creep rate
S(T ), shows an anomolous rise at intermediate temperatures associated with a drop in current
density J as shown in figure 52. The peak in the creep rate occurs at the same temperatures
that the Malozemoff plateau in S(T ) was observed (see unirradiated YBCO section) and the
value of S at the peak is of the order of six times that of the plateau (Thompson et al 1997).
Civale et al considered that at low temperatures and low fields, the vortices are individually
pinned by the columns, and vortex–vortex interactions are negligible. As the vortex density
increases the elastic interactions increase and when the elastic energy is comparable with
the pinning energy of individual tracks, collective effects take over. Bcr is the field at which
this occurs and it is temperature dependent. At low temperatures Bcr ' Bφ . According to
Bose glass theory, initial stages of relaxation should take place, via half-loop excitations.
As relaxation progresses, the size of the vortex loops become of the order of the columnar
track spacing, so that segments of the same vortex can sit on neighbouring columns. Further
relaxation should be dominated by double-kink excitation. At the intermediate temperatures
where the peak in S(T ) is observed the relaxation is anomalous in the sense that it varies nonmonotonically with time suggestive of two competing processes. Civale et al suggested that
these processes are associated with double kink excitations of individually pinned vortices
at short times and collective behaviour at longer times. Thompson et al offer a very similar
interpretation, also consistent with the Nelson and Vinokur (1992, 1993) Bose glass theory.
At B > Bφ , the peak in S(T ) is not observed.
Beauchamp et al (1995) explored Bose glass/quantum creep behaviour at millikelvin
temperatures in YBCO crystals irradiated with 605 MeV Xe ions. They found that the
relaxation rate can be divided into three regimes of behaviour depending on ratio of vortex
density to columnar defect density. Quantum creep occurs in the dilute limit, vanishing
magnetic relaxation is observed at B = Bφ in the so-called Mott insulator phase, and for
B > Bφ they observe a temperature-dependent vortex motion. Larkin and Vinokur (1995)
Magnetic behaviour of superconductors
1663
extended the original Bose glass theory to consider the dilute and dense vortex limits. Gray
et al (1996, 1997) showed that because of vortex–vortex interactions, the columns can effect
pinning at fields many times more than the matching field at low temperatures.
In the Bose glass theory for parallel columns, once a segment of vortex reaches an
adjacent column, (by thermal activation or quantum-mechanical process), the remaining
part of the vortex can follow at no additional cost. Hwa et al (1993) proposed that pinning
would be improved even further if the columnar tracks were splayed or tilted with respect
to each other. In the splayed glass phase, during the vortex hop, an ever increasing segment
of line is forced into an energetically unfavourable region. Krusin-Elbaum et al (1994a,
1996) and Schuster et al (1995a, b) (who also imaged the flux penetration into crystals using
magneto-optics), confirmed that there is a dramatic enhancement of J when the vortices
are splayed. Devereaux et al (1995) raised the issue that the misalignment of the magnetic
field and the columns may weaken the localization of the vortices and reduce the Hirr line.
4.5.5. Irradiated BSCCO 2212 crystals. Thompson et al (1992) first pointed out that the
angular selectivity seen in YBCO at high temperatures is absent in BSCCO 2212 crystals
irradiated with 580 MeV Sn ions at 20 K, although the current density and the irreversibility
line are enhanced over the unirradiated crystals. Klein et al (1993b, 1994) later showed
that in fact uniaxial enhancement is observed in irradiated BSCCO crystals, but only above
40 K. The loss of angular selectivity is either related to the fact that the system is more
two-dimensional-like at low temperatures or that random point defect rather than columnar
pinning dominates or a combination of both. Leghissa et al (1993) demonstrated loss of
translational order in columnar-defected BSCCO using high-resolution Bitter patterns.
The Hirr line was studied by Krusin-Elbaum et al (1994b) for BSCCO crystals irradiated
with 1 GeV Au ions along the c-axis. A well defined crossover field Bcr ∼ 1/2Bφ was
established. Below Bcr the Hirr ∝ (1 − T /Tc )α where α is dose dependent. Above Bcr
the Hirr (T ) line is linear. The paper also discusses the influence of columnar defects on
the melting scenario. Moshchalkov et al (1994) found close agreement with the Bose glass
theory predictions and the temperature dependence of the critical current density extracted
from the magnetization measurements. Unfortunately because these measurements were
made at remanence, the influence of self-field effects is unclear. Konczykowski et al
(1995) found giant, strongly non-logarithmic magnetic relaxation in irradiated BSCCO 2212.
The authors converted their flux creep data into I –V curves and extracted values for the
exponent µ in the interpolation formulae. (The interpolation formulae can be used because
the functional form of dependences characterizing the Bose glass phase are identical to that
for the vortex glass phase and predict power law Ueff (J ) form.) The values of µ were found
to agree with the Nelson and Vinokur predictions at 60 K and at fields much less than the
matching field. Although there is quite a bit of scatter in the data, the predicted µ = 13 for
variable range hopping was observed.
Steel et al (1996) pointed out the fact that columnar defects influence electrical properties
of Tl 2212 thin films up to fields at least 40 times that of the matching field, demonstrating
the importance of vortex–vortex interactions and also suggesting that the matching field has
no sharp significance. This is not inconsistent with Bose glass theory.
5. Summary of the questions at the brink of resolution
The complexity of the theoretical description of the vortex state has increased significantly
with the contributions from statistical mechanics produced after the discovery of the HTS
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L F Cohen and H J Jensen
(Blatter et al 1994). The balance of the competition between the three energy scales: the
vortex–vortex interaction, the vortex–pinning interaction, and the thermal energy can lead
to many very different types of behaviour.
Many of these theoretical developments are concerned with the equilibrium phases and
the nature of the transition between these phases in various model systems. As such these
theoretical developments might not be of direct relevance to experiments. One of the
problems encountered when dealing with the flux system in real superconductors is the
irreversibility, i.e. non-equilibrium features, encountered whenever pinning is relevant.
A result of the massive investigation into the properties of the flux system in HTS is
that today we have a fairly precise idea about the cardinal questions still to be completely
resolved. It is useful to distinguish between situations where the pinning energies are
negligible compared with the vortex–vortex interaction energy and the thermal energy and
the situation where the pinning energy is competing with these two energy scales.
In the case where pinning can be neglected we believe that the following list of issues
are among the most important yet to be settled and in fact are sufficiently well posed to
allow a resolution in the near future.
(1) From sharp drops in magnetization, entropy and resistance, clean untwinned YBCO
and BSCCO crystals appear to show evidence of a line of first-order transition in the
H –T phase diagram. However, theoretical concerns have raised the issue whether finitesize effects associated with a crossover from two- to three-dimensional behaviour could be
producing the semblance of a transition.
(2) In clean crystals is there always coincidence of decoupling and melting? How does
pinning influence this coincidence?
(3) Are the regions of the H –T plane fundamentally similar for YBCO and BSCCO
but occurring at different fields and temperatures refelcting the different anisotropy. Based
on the entropy change δS(T ) extracted from the magnetization jump in YBCO and BSCCO
are there fundamental differences?
(4) Is there any evidence for a line liquid?
If the temperature is low enough, the pinning originating from static disorder in the
superconducting material always becomes relevant. One of the lessons of recent research
is that the specific nature of the defects that cause the pinning is important. Point pins,
columnar pins, pinning by planes all induce very different behaviour. The following list
of questions are what we believe is the most well defined and important issues to clear up
when pinning cannot be neglected.
(5) In isotropic point disordered crystals transport critical scaling analysis appears to
show evidence for a vortex glass transition at the irreversibility line. In magnetization
measurements, the field dependences of Uc and Jc cannot easily be reconsiled with glassy
or collective pinning behaviour close to the irreversibility line measured at much lower
electric fields. This inconsistency may be related to the fact that over most of the H –T
plane, the magnetization measurement where J Jc , sets up plastic rather than elastic
behaviour. Alternatively, is it plausible that the transport measurements where J is much
closer to Jc is simply not sampling the transition effectively and cannot determine the nature
of the solid?
(6) In what way do columnar defects change the nature of the coupling between
the planes in the more anisotropic materials? How does the influence they have on the
reversible properties impact their influence on irreversible pinning behaviour? There is an
inconsistency in the way that columnar defects influence reversible properties within 1–2 K
of Tc , but do not act as effective pinning sites within 25–20 K of Tc .
(7) In the presence of correlated disorder is there sufficient evidence to prove the
Magnetic behaviour of superconductors
1665
existence of Bose glass behaviour?
(8) What is the thermodynamic equilibrium phase of the vortex system at low
temperatures in the presence of point disorder? (This question has not been addressed
by the experiments discussed in this review.)
The strive to understand the magnetic properties of the HTS has inspired an amazingly
vigourous theoretical as well as experimental line of research. Although many questions
are still open this research has been particularly fruitful in causing many new developments
in the statistical mechanics and lead to a number of beautiful experiments. Not only has
the research influenced basic science, in this way it has also laid the needed foundation
for the phenomenological understanding needed to turn the HTS into technologically useful
materials. This is a field of reseach with the potential of many new important developments
in future years.
Acknowledgments
The authors would like to thank Yuri Bugislavsky, Richard Doyle, Gary Perkins and Sasha
Zhukov for their critical reading of the text and insightful discussions. Support from the
EPSRC (LC grant no GR/K60916, and HJJ grant no GR/J36952) and from the Royal Society
are gratefully acknowledged.
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