When the magnetic field applied to a type-II superconductor exceeds
Hc1, the total free energy of the system is lowered by allowing
partial flux penetration in the form of vortices.
Since the core of a vortex line is essentially normal, there is a gain
in energy equivalent to
the condensation energy per unit length for
each vortex formed--assuming that
.However, this energy gain is more than compensated for by the
decrease in magnetic energy
per unit length
due to the region around
the vortex which is no longer diamagnetic.
The vortex line can lower its own energy by interacting with a nearby
nonsuperconducting inhomogeneity, so as to become ``pinned''.
Spatial inhomogeneities in the superconducting order parameter arising
from impurities or other structural defects, chemical vacancies,
grain boundaries, twin boundaries, etc., exert an attractive force on
the vortex. The effective range rp of the
pinning force must be at least of the order of the coherence length
(vortex core radius), since
this is the smallest length scale resolveable by the vortex core
[52]. Pinning from inhomogeneities smaller than
this is much less effective. For weakly interacting vortices, the energy saved
by the vortex line passing through a point defect of range
and length d along the vortex axis
is
(Ref. [53]). The
elementary pinning force fp acting on the vortex core is given
by
. To depin, the vortex line must move over the
distance
, so that
.Modelling extended defects, such as grain boundaries, is generally
more complicated since one must integrate over the entire inhomogeneity.
To obtain the bulk pinning force per unit volume of the
superconductor, one must sum over all the contributions from the
various pinning inhomogeneities. In general this summation is non-trivial.
In magnetic fields where the repulsive interaction between vortex lines becomes significant, the pinning of vortices to fixed positions in the superconductor can deform the vortex lattice from its ideal configuration. The deformation of the vortex lattice in response to the force exerted by a pinning center is determined by its elastic properties, namely the shear and tilt moduli c66 and c44 [54,55,56,57,58]. Deformations will increase the elastic energy of the vortex lattice. According to the ``collective pinning'' theory of Larkin and Ovchinnikov [59], the equilibrium configuration is achieved by minimizing the sum of the vortex line energy and the elastic energy of the vortex lattice. At low magnetic fields the interaction energy between vortex lines is weak, so that random pinning centers will cause only a small increase in the elastic energy of the vortex lattice. This implies that random pinning of the vortex lines will be most prominent at low magnetic fields. At high magnetic fields, weak pinning centers cannot compete with the increased strength of the vortex-vortex interactions. In this case, only strong pinning sites will hold individual vortex lines in place independently of the repulsive interaction with neighboring vortices.
In the high-Tc cuprates, the vortex lines are particularly
susceptible to pinning because the vortex lattice is ``soft''.
In particular, they have a small line tension due to
the weak coupling between the CuO2 planes which gives
way to highly flexible vortices [52].
Due to this flexibility, the vortices can become twisted, distorted
or entangled [60].
Pinning effects will be stronger in these short coherence length
superconductors.
According to Brandt [61],
randomly positioned stiff vortex lines will always broaden the SR
line shape, whereas the pinning of segments of highly flexible vortex
lines will sharpen the measured magnetic-field distribution.
In YBa2Cu3O
,
rough surfaces, oxygen vacancies and twin boundaries are the
dominant sources of pinning. In powdered samples or thin films,
pinning by rough surfaces can dominate the vortex-lattice configuration
in the bulk. Oxygen vacancies appear to be the dominant ``point-like''
defect in single crystals [53].
Twin planes occur naturally in YBa2Cu3O
along the (110) and (1
0) directions,
because of the orthorhombic crystal structure.
The depression of the order parameter at a twin boundary
attracts vortices, and can result in the creation of multivortex
chains oriented along the boundary. If the twin plane spacing is not commensurate,
this can produce distortions in the vortex-lattice geometry.
In YBa2Cu3O
, changes in the vortex-lattice geometry
can stem from a combination of twin-boundary pinning and in-plane
mass anisotropy. This will be discussed more fully below.
At low temperatures the vortices are essentially frozen into
their distorted configuration. As the temperature is raised, however,
thermal fluctuation of the vortex-line positions become important.
Thermal fluctutations in the high-Tc materials are considerably stronger
than in conventional superconductors. This is partly due to: (1) the small
value of the in-plane coherence length , (2) the high Tc
which allows for high thermal energies to be reached
in the superconducting state, and (3) the layered nature of these compounds.
Strong thermal fluctuations greatly reduce the pinning strength.
According to Feigel'man et al. [62], because of
thermal motion of the vortex lines, the vortex core
will experience a defect potential averaged over the increased
effective range
, where
is the root-mean-square (RMS)
average of the vortex-line thermal displacements
from their equilibrium positions [62].
The pinning strength is reduced by this smoothing of the effective
pinning potential accompanied by a reduction in the collective pinning force.
Thermal depinning will occur at a temperature Tp (H) at which
.The depinning of vortices results in a region
of reversibility in the phase diagram.
Below the so-called ``irreversibility line'', the vortices are pinned
by defects, whereas above this line the vortices are free to move in
response to an external force. As noted earlier, the
presence of the reversible region complicates measurements of Hc2 (T).
In particular, the resistive transition between the superconducting
and normal states is no longer sharp due to the motion of vortices
(which experience a Lorentz force from the applied current).
The energy which keeps the vortices moving is removed
from the current--so that the resistance of the material is not
zero above the irreversibility
line. Thus, it is the irreversibility line which is
usually measured, since Hc2 (T) no longer exists as a phase boundary.
If the vortex fluctuations are sufficiently large, the vortex lattice will
undergo a melting transition at a temperature Tm (H)
(<Tc) into a vortex-liquid phase. In the liquid phase, the vortex lines
are not pinned and the interaction force between vortices is weak.
As a result, there is generally no long range order in the lattice.
It is currently a matter of debate whether or not the melting temperature
Tm coincides with the thermal depinning temperature Tp.
Since pinning is sample dependent, so is the irreversibility line.
Thus, only some experiments suggest that .
Vortex-lattice melting has been observed at high temperatures
and/or magnetic fields in nearly-optimally doped, untwinned and high-quality
twinned YBa2Cu3O single crystals, from
magnetization measurements using a
mechanical torsional oscillator [63,64], from
sharp drops in resistivity measured at high magnetic fields
[65,66,67,68,69,70,71,72],
from discontinuous jumps in magnetization
measured using a SQUID magnetometer
[73,74,75,76], from
jumps in ac susceptibility measured using a Hall probe [77]
and from measured steps in specific heat
[78,79,80,81].
Many of these experiments also support a first-order melting transition
of the 3D vortex lattice in YBa2Cu3O
.
It should be noted that the melting of the vortex lattice is a phenomenon which is not unique to the high-Tc materials. Melting behaviour has been observed at high magnetic fields in Nb-Ti and Nb3Sn wires [82], polycrystalline Nb foils and NbSe2 single crystals [83] and Nb thin films [84] and Nb single crystals [85]. It should be noted that there are other more likely interpretations [86] of the measurements in Ref. [85] and other experiments [87] show no evidence for melting in Nb over the field range claimed. Recently, Ghosh et al. [88] performed AC susceptibility measurements on single crystals of NbSe2 at low magnetic fields in the vortex state. They observed a re-entrant ``peak effect'' at low fields, which may be a signature of vortex-lattice melting. The peak effect refers to an abrupt and nonmonatonic increase in the critical current density, which shows up as a negative peak in the AC susceptibility. A narrow melted-vortex region between the solid vortex state and the Meissner state was originally proposed by Nelson [89]. Figure 4.4 shows a simplified magnetic phase diagram, which roughly illustrates the vortex-solid and vortex-liquid regions.
Theoretical predictions for the shape of the melting line in the
H-T phase diagram
[62,90,91,92] are usually
based on the Lindemann criterion [93]. In this picture
the vortex lattice is expected to melt when
exceeds some small fraction cL of the intervortex spacing L. Typically
the Lindemann number cL is of the order 0.1, although experimentally,
some variation in this number is expected since the
Lindemann criterion does not account for the effects of pinning.
Pinning is expected to modify the first order melting transition, to perhaps
a ``vortex-glass'' transition [94], where the
lattice freezes into a state in which the vortices form an irregular
disordered pattern or into a highly
disordered state in which the vortex lines are ``entangled''
[95,96].
The melting transition in the H-T phase diagram is reasonably described
by the power-law relation
in moderate
magnetic fields
.Brandt [91] and Houghton et al. [92] considered a
nonlocal elastic theory for the vortex lattice to arrive
at a power-law exponent
.
Blatter and Ivlev [97,98] later argued that this result is really
only valid in YBa2Cu3O
close to Tc. They performed a more rigorous
calculation which takes into account the suppression of the order
parameter near Hc2(T), as well as quantum fluctuations, to yield a melting line which is
better described with a smaller value of n. This prediction is supported by several
experiments on YBa2Cu3O
which report exponents of
[65,68,70,73,75,76,77,99].
Some of these experiments [75,76,77] report power-law
dependences for the melting line in which
,
the critical exponent expected within the 3D XY critical regime [100].
Although YBa2Cu3O is a layered material, near optimal doping
the vortex lattice behaves in an essentially three-dimensional manner
over most of the H-T phase diagram.
This is not the case for the highly anisotropic compound
Bi2Sr2CaCu2O
, where the coupling between planes
is very weak even well below Tc. For this material it is useful
to consider the 3D vortex line as being composed of a stack of
aligned 2D vortex ``pancakes'', where the pancakes exist within the
superconducting layers (i.e. CuO2 planes) [101].
The Lawrence-Doniach (LD) model [102] is a reasonable
starting point
for a theoretical treatment of this problem. In this model adjacent
superconducting layers are separated by an insulating layer of thickness s.
The vortex pancakes in neighboring layers are connected by Josephson vortices
which exist within the Josephson junctions between the superconducting
layers. The vortex pancakes in adjacent CuO2 planes thus couple
through both magnetic interactions and
Josephson tunneling. A third coupling mechanism, namely the indirect
effect of the Coulomb interaction, has been suggested by Duan [103].
The relevant parameter in the LD model
is the ratio of the
-axis coherence length
to s.
When
there is no phase difference in the order parameter
between neighboring layers, so that in the absence of pinning the vortex lattice
exhibits 3D behaviour--equivalent to the anisotropic London
and GL models. On the other hand, when
there
may be a
phase difference and the LD theory describes a quasi-2D vortex structure.
The LD model will not be completely satisfactory in a superconductor
in which the material between the superconducting
layers is not completely insulating. In this case the
proximity effect may become important.
At low temperatures vortex pancakes between neighboring layers are aligned.
However, in a superconductor with random inhomogeneities, pinning will
displace some of the pancakes and cause a suppression of the phase
coherence between layers [104]. The effects of
random pinning-induced misalignment of the vortex pancakes on
the measured SR field distribution has been the focus of
several studies
[61,104,105,106,107,108,109].
The effects include a reduction in both the line shape width
and the line shape asymmetry. When the magnetic field is increased,
the interaction between pancake vortices within a layer will eventually
exceed the coupling strength between the pancake vortices in neighboring
layers. In this case random pinning in the layers will lead to a
misalignment of the pancake vortices between layers.
Thus, in a highly anisotropic system with inhomogeneities,
a dimensional crossover
from a 3D to a 2D vortex structure can be induced by magnetic field.
Harshman et al. [106] observed a narrowing and a loss
of asymmetry in the
SR line shape for Bi2Sr2CaCu2O
at low temperatures and high magnetic fields, which they attributed
to pinning-induced misalignment of the pancake vortices. In the same
study, the
SR line shape for YBa2Cu3O
under similar
conditions was found to be in agreement with a 3D vortex lattice.
Other
SR studies on Bi2Sr2CaCu2O
[107,108] provide additional support
for a field-induced dimensional crossover.
Clem [101] has shown that the thermal energy required to misalign 2D pancake vortices is extremely small. The effect of thermal fluctutations on the vortex lattice is very different between the regions of weak and strong magnetic fields [110]. In low magnetic fields the displacement amplitude of the pancake vortices due to thermal fluctuations is much larger than the relative displacement of the vortices between layers. On the other hand, as just noted, in strong magnetic fields the vortex-vortex interactions within a layer are stronger than those between layers. In this case thermal fluctutations act in a quasi-2D manner.
The effects of thermal fluctuations on the measured internal field
distribution have been previously studied by SR in
Bi2Sr2CaCu2O
[107,111].
Rapid fluctutation of a vortex about its average
position can increase the apparent core radius and smear the magnetic field
out over an effective radius of
[61].
The smearing effect reduces the average of the field distribution
in the vortex-core region.
The muon detects the field averaged over the fluctutations, since
the typical time scale
for thermal fluctuations of the vortices (
s [51])
is much shorter than
, where
is the muon gyromagnetic ratio and
is the range of the field
fluctuation at the muon site.
The result is a premature truncation of the high-field tail
in the measured
SR line shape. A proper analysis of the corresponding
muon precession signal would lead to an overestimate of the vortex-core
radius r0.
The effect of thermal fluctuations on the high-field tail was
nicely demonstrated in Ref. [111].
The melting transition in Bi2Sr2CaCu2O
was determined by Lee et al. [107,111] by
observing additional changes in the
SR line shape--namely,
a reduction in the line width and in the asymmetry of the line shape as
a function of temperature and magnetic field.
Numerical simulations of the magnetic field distribution
were later provided by Schneider et al. [112],
for both a vortex liquid phase and a disorder-induced 2D phase.
Good agreement was reported between these theoretical line shapes
and those measured in the experiments by Lee et al.
Although the coupling strength between CuO2 planes in fully oxygenated
YBa2Cu3O is sufficient to yield a vortex structure
which exhibits 3D behaviour over the majority of the
H-T phase diagram, such is not the case in the underdoped
material. Magnetization measurements performed on YBa2Cu3O6.60
are consistent with quasi-2D fluctutation behaviour [113].
As I will show later, due to this reduced dimensionality,
SR measurements of the internal magnetic
field distribution in YBa2Cu3O6.60 yield a rich phase
diagram which is comparable to that
for Bi2Sr2CaCu2O
.