The problem of an isolated vortex line in a *d*_{x2-y2}-wave
superconductor was first seriously considered by
Soininen *et al.* [166], using a simple microscopic
model for electrons on a lattice in the BdG formalism.
In calculating the spatial distribution of the order parameter
for a single vortex, they found that
an *s*-wave component is induced near the vortex core with opposite
winding of phase relative to the *d*-wave component. Several authors
[167,168,169,170,171] have
studied the effect of this induced *s*-wave order parameter on both an isolated
vortex and the vortex-lattice structure, in terms of
two-component GL equations containing
both *s*-wave and *d*-wave order parameters. In these equations the *s*-wave
component couples to the *d*-wave component through mixed gradient terms.
Because of this coupling, the *s*-wave component is induced by spatial
variations in the *d*-wave order parameter which occur in
the vicinity of a vortex line. In a tetragonal superconductor
the induced *s*-wave order parameter has fourfold symmetry and the
*d*-wave order parameter has circular symmetry. Thus,
in the core region of an isolated vortex, the magnetic field distribution
is fourfold symmetric, whereas away from the core region, where the
*s*-wave component vanishes, the field distribution
has circular symmetry. At low
temperatures near *H*_{c2}, the vortex lattice is oblique--reflecting
the fourfold symmetry of the *s*-wave order parameter.
However, near *T*_{c} the *s*-wave component
becomes negligible and the vortex lattice is triangular.
This latter prediction is crucial to the study of
YBa_{2}Cu_{3}O in this thesis.
Later it will be shown that when
YBa_{2}Cu_{3}O_{6.95} is cooled through *T*_{c} in the presence
of a magnetic field, the vortex lattice becomes strongly pinned at
and remains so for further reductions
in temperature. Thus the vortex lattice geometry at low temperatures
is governed by the geometry at the pinning temperature--which is
nearly triangular in the two-component GL model.

It is well known that the orthorhombic crystal structure of
YBa_{2}Cu_{3}O results in a significant
mass anisotropy in the -
plane--although the actual value of is clearly a doping-dependent quantity.
For instance, according to infrared reflectance measurements at zero field:
and Å in YBa_{2}Cu_{3}O_{6.95}
[172], while
and
Å
in YBa_{2}Cu_{3}O_{6.60} single crystals [173]
similar to those used in the present study.
Xu *et al.* [170] have extended the two-component GL theory
to include the effects of mass anisotropy. When the magnetic
field is applied parallel to the crystallographic -axis,
both the *s*-wave and *d*-wave order parameters show a two-fold
symmetry, where the *d*-wave order parameter has essentially elliptical
symmetry. Within the GL formalism, Heeb *et al.* [174]
find a similar reduction from fourfold to twofold symmetry
in the presence of orthorhombic distortions.
More recently, Ichioka *et al.* [175] reconstructed the
two-component GL theory to investigate the vortex lattice
in a pure *d*_{x2-y2}-wave superconductor at low temperatures near *H*_{c2}.
These authors argue that correction terms derived from the Gor'kov equations
which are absent in conventional GL theory must be included
at low *T*. They find that the unit-cell shape of the vortex lattice
transforms from hexagonal to square at low temperatures, with the
fourfold symmetry of the cores becoming clearer, even when there is
no induced *s*-wave component included in the theory.

It is important to realize that the results
using the two-component GL theory are strictly valid only
along the superconducting-to-normal phase boundary
near *H*_{c2}, and therefore do not necessarily provide an
understanding of the vortex-lattice structure deep in the superconducting
state where SR experiments are performed.
Near *H*_{c2}, where the vortices are close together,
the fourfold symmetry of the induced *s*-wave component in the cores
leads to a fourfold symmetry in the vortex-lattice configuration.
However, there is no reason to expect this to be the case
at low fields where the density of vortices in the superconductor
(and hence the influence of the cores on the flux-lattice geometry)
is diminished.
For instance, in the borocarbide superconductors
RNi_{2}Bi_{2}C (R= Er, Lu) the vortex lattice has been shown to transform
from square to triangular at low fields [176,177].
Although the origin of the fourfold symmetry at high fields
is as yet unresolved in this family of compounds, it is clear
that the geometry of the vortex lattice can change with reduced
vortex-vortex interactions. Of course all of this is irrelevant if the
vortex lattice ``freezes'' in at high temperatures, as mentioned earlier.

Another serious problem with the two-component GL theory, is that it
contains too many phenomenological
parameters to be useful in fitting the measured internal field distributions.
Recently, Affleck *et al.*
[178] attempted to resolve these issues by generalizing
the London model to include four-fold anisotropies in a tetragonal material.
Starting from a GL free energy density with *s* and *d*-wave order parameters,
they derived the corresponding London equation.
For a magnetic field applied along the -axis,
the field profile which is obtained may be written as [178]

Franz *et al.* [179] have recently developed a generalized
London model derived from a simple microscopic model, which takes into
account the nonlocal behaviour which occurs in the vicinity of the nodes
in a *d*_{x2-y2}-wave superconductor. This modified London model predicts
novel changes in the vortex-lattice geometry, including
two first order phase transitions at low *T*.
More recently, this work has been extended to account for both
nonlinear and nonlocal effects as discussed earlier [40].
It is found that the nonlocal corrections are the dominant effect
in determining the vortex-lattice geometry. In particular, the numerical
calculations in Ref. [40] yield a nearly triangular vortex
lattice. It should be noted, however, that the source
term the authors used in the London equation was derived from the GL
equations near *H*_{c2} [120], and is not theoretically valid
for lower magnetic fields.

Shiraishi *et al.* [180] have studied the
vortex lattice using the extended GL theory, which includes the
fourth-order derivative term and accounts for the finite size of the
vortex cores. The fourfold symmetry of the vortex cores leads to
a first order transition in the vortex-lattice geometry
with increasing magnetic field.
In particular, in weak fields the vortex lines form a triangular lattice
which slowly transforms with increasing magnetic field, and then suddenly
changes to a square lattice. Near *T*_{c} they predict a crossover field
given by .

The structure of a single isolated vortex
in a pure *d*_{x2-y2}-wave
superconductor has been calculated using the quasiclassical Eilenberger
theory [181,182], which is valid at arbitrary temperature.
A fourfold symmetry appears in the LDOS, the pair potential,
the supercurrent and the magnetic field distribution around a vortex.
The fourfold symmetry about the vortex center
is strongest in the core region and gradually fades to circular symmetry
toward the outer region. On the other hand, using an approximate
version of the BdG
equations Morita *et al.* [183] found that the LDOS around
a single *d*_{x2-y2}-wave vortex has circular symmetry, and exhibits
fourfold symmetry only when an *s*-wave component is mixed in.
Franz and Ichioka [184] have since argued
that the circular symmetry obtained by these authors is an
unphysical artifact of the approximations used for the BdG equations.
The BdG equations have been solved numerically for a
vortex lattice of a *d*-wave superconductor [166,185].
Unfortunately, there are currently no calculations (in any formalism)
of the vortex-lattice structure in a *d*_{x2-y2}-wave superconductor,
which are
valid at both low *T* and low *H* where experiments are generally performed.

Several authors [182,185,186] have suggested
that the low-lying
quasiparticle excitations cannot be bound in a *d*-wave vortex core
because of the presence of the nodes. Rather than states which are localized
in the core as in a *s*-wave superconductor, the states are peaked in
the core region but extend along the node directions.
According to Ichioka *et al.* [182], in the vortex state
of a *d*_{x2-y2}-wave superconductor the quasiparticles do not flow along
conventional closed circular trajectories, but rather flow along
open trajectories which connect with those of nearest-neighbor vortices.
This theoretical model requires that the nodes lie along the
line connecting nearest-neighbor vortices. In the absence of
anisotropy, this implies that the nearest-neighbor direction must be
45.

So far, experiments have not entirely resolved the issue of
the vortex-lattice structure in the high-*T*_{c} materials either.
The major problem
has been in determining how much of the observed vortex structure is
directly attributable to the *d*_{x2-y2}-wave pairing state
and how much is due to deformations of the lattice caused by extrinsic
effects. Generally speaking, Bitter decoration
experiments which image the vortex lattice at the sample surface
indicate that the vortices arrange themselves to form a triangular
lattice. For instance, Gammel *et al.* [187]
observed a triangular lattice with long-range order in
YBa_{2}Cu_{3}O_{7}. Decoration experiments by
Dolan *et al.* [188] show a
triangular lattice in YBa_{2}Cu_{3}O_{7}, with a slight distortion probably
caused by the - plane anisotropy.
A triangular vortex lattice with long-range order was also observed by
Vinnikov *et al.* [189] in Tl_{2}Ba_{2}CaCu_{2}O_{x}
at and in Bi_{2}Sr_{2}CaCu_{2}Oby Kim *et al.* [190] at high temperatures.
Since the Bitter decoration technique is resolution
limited to low magnetic fields, the results obtained
may not be representative of the lattice structure at higher
magnetic fields, particularly in a sample dominated by extrinsic effects.

The structure of the vortex lattice in the bulk of a superconductor
can be investigated using small-angle neutron scattering (SANS).
The pattern generated by neutrons scattering from the vortex lattice
is the reciprocal lattice of the real-space vortex lattice.
Large single crystals are generally required so that the diffracted
neutron intensity is strong enough to clearly resolve the peaks
resulting from Bragg reflection.
The scattered intensity is proportional to the square of the
spatial variation in the local
magnetic field, which is of the order [191].
In principle, one can measure the temperature dependence of using SANS. The signal-to-noise ratio is reduced in samples which
contain defects which scatter the neutrons in the same small angle
as those scattered from the vortex lattice. In clean conventional
superconductors like Nb and NbSe_{2},
a perfect triangular lattice with long-range order is observed
using SANS [165,192].
A triangular lattice has also been observed in
the high-*T*_{c} superconductor Bi_{2}Sr_{2}CaCu_{2}Oat low temperatures [193].
On the other hand,
resolving the vortex-lattice geometry in YBa_{2}Cu_{3}Ohas been more difficult.
Forgan *et al.* [191] investigated the vortex lattice
in small single crystals of YBa_{2}Cu_{3}O_{7} up to fields of 0.6 T.
Only diffraction spots corresponding to vortices parallel to the
twin boundaries were strong.
Relatively weak diffraction spots were observed in the other directions--which
implies that the vortex lattice was non-uniform.
A diffraction pattern with square symmetry was observed by Yethiraj *
et al.* [194] in a SANS study of the vortex
lattice in YBa_{2}Cu_{3}O_{7}. The authors attribute the observed geometry
to twin planes, since the intensity peaks are aligned along the (110)
direction.
More recently,
Keimer *et al.* [195] studied the vortex lattice in
a larger single crystal of YBa_{2}Cu_{3}O_{7} for magnetic fields of
0.5 to 1.5 T applied along the crystallographic -axis,
at K.
These authors reported that the vortices form an
oblique (fourfold symmetric) lattice
with an angle of between two nearly equal primitive vectors, and
that one of the primitive vectors is
oriented at an angle of with respect to either the
or axis. The alignment of one primitive vector of
the oblique lattice
with the (110) or (10) direction of the underlying crystal lattice
was observed in four different orientational domains of the crystal.
Walker and Timusk [196] noted that the vortex-lattice
geometry observed in this SANS experiment is easily explained as a
combination of strong pinning effects due to twin planes and
the - plane anisotropy in YBa_{2}Cu_{3}O_{7}.
In particular, an equilateral-triangular vortex lattice with one side
aligned along a twin boundary, which is then stretched
(due to the - plane anisotropy)
along a line which makes a angle with the twin
boundary, yields the observed vortex-lattice geometry.

An oblique lattice was also found in STM measurements
performed by Maggio-Aprile *et al.* [197]
on the (001) surface of twinned
YBa_{2}Cu_{3}O, at T and K.
Consistent with the SANS results,
they report an angle of between nearly equal primitive
vectors--although they could not determine the orientation
of the vortex lattice with respect to the crystal lattice.
The oblique lattice imaged in this experiment was only observed locally,
with no apparent long-range order.
These authors also report that the vortex cores are ellipsoidal
in shape with the ratio of the principle axes being about 1.5.
The elongation of the cores is consistent with the
- plane anisotropy in YBa_{2}Cu_{3}O--*i.e.*
other than this anisotropy
the vortex cores appear to be approximately circular.
As is the case for the SANS experiments, the geometry of the observed
vortex lattice can also be explained as a combination of the
- plane anisotropy
and an alignment of vortex lines with twin boundaries. Thus in a detwinned
or sparsely twinned sample of YBa_{2}Cu_{3}O, it is likely
that the vortex lattice is triangular at moderate
magnetic fields.

There are several serious discrepancies between the current experiments on
high-*T*_{c} superconductors in the vortex state and the theoretical models
for the vortex lattice of a *d*_{x2-y2}-wave superconductor:

- 1.
- None of the current theoretical models can explain, in terms
of an intrinsic mechanism, both the vortex-lattice geometry
(observed in the SANS and STM experiments on YBa
_{2}Cu_{3}O) and the orientation of the vortex lattice with respect to the crystallographic axis (observed in SANS experiments in YBa_{2}Cu_{3}O). To fully understand the influence that the symmetry of the pairing state has on the vortex-lattice geometry, it will be necessary to perform imaging experiments on untwinned crystals and/or tetragonal superconductors. - 2.
- The STM image of the vortex core on
YBa
_{2}Cu_{3}O by Maggio-Aprile*et al.*[197] does not show the fourfold anisotropy predicted for the LDOS near a*d*_{x2-y2}-wave vortex. - 3.
- Experiments performed on both YBa
_{2}Cu_{3}O [197,198] and Nd_{1.85}Ce_{0.15}CuO[199] are more consistent with a picture in which a few bound quasiparticle states exist in the vortex core--which contradicts the idea that the low-lying quasiparticle excitations cannot be bound in a*d*_{x2-y2}-wave vortex core because of the nodes. For instance, in the STM experiment on YBa_{2}Cu_{3}O [197] two peaks separated by about 11 meV were observed in the differential conductance*dI*/*dV*(*i.e.*the LDOS) measured at the center of a vortex core. A natural interpretation of this result is that these peaks correspond to the lowest bound quasiparticle energy levels.

Franz and Tesanovic [186] have recently attempted
to address some of these issues by proposing a mixed *d*_{x2-y2}+*id*_{xy}
pairing state. For this symmetry there is a finite energy gap
everywhere at the Fermi surface so that bound quasiparticle
states can exist. Within a BdG formalism these authors predict
near spatially-isotropic bound quasiparticle states--however,
the size of the *d*_{xy} component required to reproduce the
gap observed in the tunneling conductance at the center of the
vortex core in Ref. [197] may be difficult to
reconcile with other experiments. For instance, the linear-*T*
dependence of found in
our own SR studies of YBa_{2}Cu_{3}O in
the vortex state
[2,3,5] imply that any finite gap
which opens along the node directions cannot be too large.

Given the anomalous normal state properties in the high-*T*_{c}
compounds, it is quite reasonable to expect deviations from the
conventional picture of a vortex core in these materials.
One such novel description is the prediction from *SO*(5) theory
for the existence of a superconducting vortex
with an antiferromagnetic core in underdoped high-*T*_{c} compounds
[200,201]. Recently, we have investigated this
possibility in a SR study of YBa_{2}Cu_{3}O_{6.57}
(Ref. [202]). However, neither this study or any other has
produced clear evidence for the existence of antiferromagnetism
in the vortex core.

With experiment and theory in apparent disagreement, it seems
reasonable to model the vortex lattice in the simplest possible
manner. Furthermore, it does not matter whether theory predicts
a fourfold symmetric vortex lattice for YBa_{2}Cu_{3}Oif pinning ``freezes in'' the threefold (triangular) lattice
just below *T*_{c}.
In this thesis,
the contribution of the vortex lattice to the
measured SR line shape will be modelled with both Brandt's
modified London model [see Eq. (4.10)] and Yaouanc's version of Hao's
analytic solution to the GL equations [see Eq. (4.13)].
The primary advantage of using these phenomenological models is that
they contain a manageable number of parameters for fitting the SR spectra.
In addition, both models are conveniently valid
at low reduced fields *b* where SR experiments are generally performed.
I will show in this study that fitting to these models yields
a good description of the *T* and *H*-dependences of
the fundamental length scales and (*i.e.* *r _{0}*)
in both conventional
and unconventional superconductors.