In this chapter, recent SR measurements of the
-
plane magnetic penetration depth
and the vortex core size (
)in the high-Tc superconductor YBa2Cu3O6.60
are presented.
Figure 8.1 shows the SR line shapes for the underdoped compound
YBa2Cu3O6.60 (U1) and the optimally doped compound
YBa2Cu3O6.95 (O1), at similar temperature and
magnetic field. The width of the field distribution in the underdoped
compound is considerably smaller due to a larger
.Furthermore, the high-field cutoff is much more pronounced due to both
a larger vortex-core radius and the longer
.
In the analysis that follows, it is found
that at
T the fractional volume of the sample occupied
by the vortex cores in YBa2Cu3O6.60
is nearly two times greater than in YBa2Cu3O6.95.
Figure 8.2 shows what happens to the internal field distribution
in YBa2Cu3O6.60 upon warming the sample in an applied field
of 1.49 T. In going from (see Fig. 8.1) to
(i.e. 24.8 K), the asymmetry of the
line shape suggests that the vortex lattice is comprised of
3D-flux lines arranged in a regular pattern. However, at
the line shape becomes more symmetric, and is completely
symmetric at
. The loss of asymmetry in
the
SR line shape is a strong indication that the vortex lattice
has melted and/or it has undergone a 3D-to-2D transition.
In order for
and
to be determined in
YBa2Cu3O6.60 using
SR spectroscopy, a rigid 3D-vortex
lattice must exist. We will therefore begin by restricting ourselves
to the low-temperature region of the H-T phase diagram,
and reserve the discussion of the vortex lattice in
YBa2Cu3O6.60 at higher temperatures
till the end of this section.
Due to the larger contribution of the vortex cores to the measured
internal field distribution, it is unnecessary to
relate or fix any of the parameters in the fitting procedure.
In this way the data is
analyzed exactly as it was for NbSe2. We note that the theoretical
field distribution is generated assuming a triangular vortex lattice.
This is reasonable, since we know of no experiments which have imaged
the vortex lattice in YBa2Cu3O6.60.
As was explained for
YBa2Cu3O6.95, in the absence of twin boundary pinning
the -
plane anisotropy will stretch the triangular lattice
in a way which does not change the shape of the internal field distribution.
Figure 8.3 shows the temperature
dependence of the fitted
in
YBa2Cu3O6.60 (U1) using the analytical GL model
for the field profile of the vortex lattice. The curves through the
data points are merely guides for the eye. At higher temperatures
where the vortex lattice is no longer a rigid 3D structure, the
fitted value of
merely characterizes the width of
the field distribution and is not the penetration depth as defined
at lower temperatures. Note that the transition in the vortex-lattice
structure occurs at a lower temperature in the larger magnetic field.
Figure 8.3 suggests that measurements of
should be
restricted to temperatures below 25 K for applied magnetic fields
greater than 1.5 T.
Figure 8.4 shows the temperature dependence of in YBa2Cu3O6.60 (U2) at low T, for three of the magnetic
fields considered. As in YBa2Cu3O6.95, there is a strong linear
decrease in the superfluid fraction with increasing
temperature. The solid lines are a fit to Eq. (7.2).
The fitted parameters appear in Table 8.6 along with those
for an analysis assuming the ML model with a Gaussian cutoff factor.
Note that the coefficient
of the term linear in t
is weaker than in the optimally
doped compound and is at most only weakly dependent on magnetic field.
This is consistent with conclusions drawn from
earlier
SR measurements of the muon depolarization rate
for various oxygen dopings in YBa2Cu3O
[209].
As shown in the inset of Fig. 8.4, there is also excellent agreement
with microwave cavity measurements of
K) in
zero magnetic field [210]. According to the Meissner-phase
measurements in Ref. [210], the strength of the term
linear in t
changes substantially as a function of oxygen doping
only in the
-direction
(i.e. the direction of the CuO chains). In particular, the
value of the coefficient linear in t
for the temperature dependence
of
is reduced in the underdoped compound. One interpretation
is that the oxygen vacancies in the CuO chains act as
pair-breaking defects.
2c|![]() |
2c|![]() |
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Magnetic | Beamline/Year | ||||
Field | ![]() |
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|
[T] | [Å] | [10-1] | [Å] | [10-1] | |
1|l|(i) | |||||
---|---|---|---|---|---|
0.742 | m15/1994 | 2212(18) | 4.1(2) | 1698(15) | 4.9(2) |
1.013 | m15/1994 | 2248(17) | 4.0(3) | 1739(23) | 4.7(2) |
1.490 | m15/1994 | 2393(40) | 4.5(6) | 1810(32) | 5.7(8) |
1|l|(ii) | |||||
0.500 | m15/1996 | 2286(11) | 4.1(2) | 1754(13) | 4.6(2) |
0.850 | m15/1996 | 2396(18) | 4.1(2) | 1815(16) | 5.7(3) |
1.250 | m15/1996 | 2450(19) | 3.9(4) | 1846(17) | 4.8(4) |
The magnetic-field dependence of extrapolated to
is shown in Fig. 8.5. There is a clear difference
in the magnitude of
determined from measurements
in the twinned (U1) and detwinned samples (U2).
This difference is likely due to vortex-lattice distortions in U1
caused by twin boundary pinning of the vortex lines. Such
distortions introduce a systematic uncertainty in the determination
of
. As noted earlier, fitting to a theoretical
field distribution which assumes the wrong vortex-lattice geometry
seriously affects the magnitude of the extracted
, but
has little effect on the T or H-dependence.
The lines in Fig. 8.5 are fits to Eq. (7.3) assuming
. The fitted parameters are:
Å and
Å/T
in the twinned sample (U1) and
Å and
Å/T
in the detwinned sample (U2).
To roughly estimate Hc2(0) in YBa2Cu3O6.60, we can use
the ratio of the coherence length between this compound and
YBa2Cu3O6.95. It was noted earlier, that at
T
the fractional volume of the sample occupied by the vortex cores is
nearly two times greater in the underdoped compound. This implies that
in YBa2Cu3O6.60 is
times the coherence length in YBa2Cu3O6.95.
Taking
T
for YBa2Cu3O6.95 and assuming
,we may estimate
T in YBa2Cu3O6.60.
Using Hc2 (0) in place of Hc2 (0.33 Tc) in Eq. (7.4),
the coefficient linear in h (i.e
) at
is
5.6 and 4.2 in U1 and U2, respectively. On the
other hand, for the YBa2Cu3O6.95 compound treated earlier,
in both the O1 and O2 samples.
The linear dependence of
on H
when plotted as a function of reduced field is therefore smaller in
YBa2Cu3O6.60 than in YBa2Cu3O6.95.
It is predicted that the presence of
impurities should reduce the field dependence of the penetration
depth in the nonlinear supercurrent response of a dx2-y2-wave
superconductor [36]. It appears as though
oxygen vacancies in the chains of the underdoped compound
have a similar effect.
In the analysis of the SR spectra for the optimally doped compound
it was assumed that
. As shown
in Fig. 8.6, fits to the underdoped compound without this constraint
reveal that this is a reasonable approximation. Moreover, the range
of values of
is small, so that the uncertainty introduced
by this assumption into the analysis of the
YBa2Cu3O6.95 data is also small.
The variation of the additional
broadening parameter with temperature in
YBa2Cu3O6.60 (U2) is shown in Fig. 8.7(a)
for
T (open circles) and
T (solid circles).
The magnitude of
is nearly the same at the two
magnetic fields shown.
As in the optimally doped compound,
there is a reduction in the value of
with increasing temperature
and the quality of the fits are independent
of temperature and magnetic field [see Fig. 8.7(b)].
Figure 8.8(a) shows the temperature dependence of
calculated from
at these fields.
The field dependence of
is weaker in
YBa2Cu3O6.60 than in the optimally doped
compound. When the
RMS displacement of the vortex lines from their ideal positions
is expressed as a percentage of the intervortex spacing L
[see Fig 8.8(b)],
the percent disorder at
T agrees well with that
for YBa2Cu3O6.95 [see Fig 7.11(b)].
However, the percent disorder is actually slightly larger at
T
in the underdoped compound.
This may reflect the quasi-2D nature of the vortex lines in
YBa2Cu3O6.60. In particular, there is likely some degree
of pinning-induced misalignment between the 2D vortex lattices in adjacent
layers at the stronger field, so that the vortex lines are no longer
straight.
Similar values of
and
were found in the twinned sample U1.
Figure 8.9 shows
the temperature dependence of the GL parameter in YBa2Cu3O6.60 (U2) at three magnetic fields. There is far less
scatter in the data at the higher fields compared to the optimally
doped samples (note that the vertical scale is larger than
in Fig. 7.12).
This is due to the increased size of the vortex cores in the underdoped
compound over the full field range considered.
The horizontal lines in
Fig. 8.9 are
fits to a T-independent
. The field dependence of
obtained from these fits and those for the twinned sample (U1)
is shown in Fig. 8.10. The values of
in the
twinned and detwinned
samples agree well enough that a single fit to Eq. (7.5)
was performed.
The fit gives
which is larger than in
YBa2Cu3O6.95, and
[assuming
T],
which is smaller than
in YBa2Cu3O6.95 but still considerably greater than in NbSe2.
The larger value of
results from the substantial increase in
the penetration depth in the CuO chain direction,
whereas the reduced value of the coefficient linear in h stems
from the weaker field dependence of
.
Figure 8.11(a) shows as a function of temperature
in YBa2Cu3O6.60 (U2), whereas
Fig. 8.11(b) shows the
temperature dependence of r0 obtained from the Js (r) profiles.
The net change in
over the
temperature range
to
K is greater than
in YBa2Cu3O6.95, although when
is plotted
as a function of reduced temperature, the term linear in t is weaker.
For instance, fitting to the linear relation
![]() |
(1) |
The values of and r0(H) extrapolated to
are
shown in Fig. 8.12 for both the
twinned (U1) and detwinned (U2) samples.
Note that there is good agreement between both samples, which implies
that the twin planes have little effect on the vortex-core size. The
magnetic field dependence of
in
YBa2Cu3O6.60 is shown compared to that
in YBa2Cu3O6.95 in Fig. 8.13. The coherence length in
the vortex state is larger in the underdoped material at all fields
considered. The solid curve through the data points of the
optimally doped sample was described in the previous section.
The solid curve through the
data points for YBa2Cu3O6.60 was obtained by averaging the
linear best-fit lines for
and
of samples
U1 and U2, so that
.As was explained in the discussion for NbSe2, the curves in
Fig. 8.13 should flatten out at low fields, when
(i.e. there should be no change in the vortex
core radius when the vortices are isolated from each other).
From our
SR results, the coherence length should stop rising below
in YBa2Cu3O6.95 and below
in YBa2Cu3O6.60.
Let us now return to the discussion of vortex-lattice melting
in YBa2Cu3O6.60, which occurs
well below the superconducting-to-normal phase transition.
As noted earlier, the sudden loss of asymmetry in the SR line shape
as the temperature is raised may be interpreted
as a melting transition for the vortex lattice (see Fig. 8.2).
In the
melted phase the vortices are mobile, giving rise to a vortex
structure with no long range spatial order.
As indicated in Fig. 8.3, the melting temperature
Tm depends strongly on magnetic field.
By monitoring the changes in shape of the
measured field distribution with increasing temperature, at several different
magnetic fields, we find that the melting transition can be described by
![]() |
(2) |
The dimensionality of the ``melted'' vortex lattice is of equal interest here. The question is whether the ``melted'' lattice consists of straight 3D-vortex lines, highly flexible 3D-vortex lines or pancake vortices in which the phase coherence is destroyed across neighboring CuO2 planes. The strong magnetic field dependence of Tm favours the latter scenario. Within the LD model, the pancake vortices are aligned and weakly coupled between neigboring CuO2 layers at low temperatures in the absence of disorder. As the temperature is increased, thermal fluctutations of the strings of pancake vortices becomes significant. The displacement of the 2D-pancake vortices within a layer by thermal fluctuations is opposed by the strength of the Josephson-coupling between pancakes in neighboring CuO2 layers. At low magnetic fields, the vortex-vortex interactions within a layer are weak, so that the displacement of an individual vortex does not necessarily affect the other vortices in the layer. Thus, for small thermal fluctuations a 3D structure is maintained in the majority of the sample. It has been suggested that at low fields, the vortex lattice first melts into a state consisting of mobile 3D vortex lines, which are destroyed upon further increase in the temperature as the pancakes which makeup the vortex lines breakup [110].
On the other hand, at high magnetic fields, the
interaction between pancake vortices within a CuO2 layer
is stronger than the interaction between pancake vortices between
neighboring layers. In this case, the vortex lattice behaves in a
quasi-2D manner, consisting of weakly interacting 2D-vortex lattices
in different CuO2 layers. Thermal fluctuations cause the 2D lattices
to move back and forth over top of each other. For larger fluctuations
the pancakes within the layers become mobile.
The effect of temperature and magnetic field
on the SR line shape
in YBa2Cu3O6.60 is similar
to that previously observed in
SR experiments on
Bi2Sr2CaCu2O
[107,111] and arises here
due to a reduction in the
-axis coherence length relative to that
in the optimally doped compound YBa2Cu3O6.95.
Pinning will of course also lead to vortex displacements. As discussed
earlier, the effects of pinning on the vortex structure in the low
and high-field regimes will be considerably different in the
pancake-vortex model. Figure 8.14 shows
the SR line shape in ``twinned'' YBa2Cu3O6.60 (U1),
obtained by cooling the sample through the transition to low temperature in
different magnetic fields.
The field distributions at
T and
T
exhibit the characteristic features of a 3D-ordered vortex lattice.
However, the field distribution which arises from field cooling
in a magnetic field of 2.91 T does not exhibit
the expected high-field tail for a 3D-ordered vortex lattice, but rather
shows a low-field tail.
From the melting-line expression obtained above, the lattice should
melt at
K at
T, which is
higher than
K in Fig. 8.14.
The most natural interpretation is that the 2D-vortex lattices become
misaligned with respect to each other due to pinning brought about by
pointlike defects within the layers. The origin of the low-field tail
in Bi2Sr2CaCu2O
has been previously attributed
to the geometry of the sample [112,211,212].
In a flat rectangular sample, the average magnetic field penetrating near the
crystal edges is smaller than that in the center of the sample,
due to non-uniform demagnetization. As a
result, the density of vortices is smaller near the sample edges.
The sample-geometry effect of course also exists when the vortex
lattice is 3D and ordered. However, the variations of
the internal field distribution due to the vortex lattice ``swamp''
out the effect. Also, in the case of
YBa2Cu3O6.95 and at low fields in YBa2Cu3O6.60,
the high pinning temperature likely freezes in
a nearly uniform density of vortices in the sample, which remains uniform
upon cooling.
A tiny low-field tail is visible in all of the line shapes
shown earlier.
The effect of the sample geometry on the measured internal field distribution
can occur without vortex pinning--as evidenced by the existence of a
large low-field tail in the SR line shape for
Bi2Sr2CaCu2O
in the melted phase [107].
The top panel of Figure 8.15 shows
the
SR line shape measured in the
twinned sample of YBa2Cu3O6.60 (U1) obtained by field
cooling in a magnetic field
T to
K,
followed by an increase of approximately 0.01 T in the applied field.
Note that the small background signal shifts to the new applied field at
T, but the signal from the sample does not.
This verifies that the vortex lattice is not melted and strongly
supports the picture of a dimensional transition which is induced at
high magnetic fields in the presence of pinning.
When the temperature is raised, the 2D-vortex lattices depin within
the layers and the lattice melts (see middle and bottom panels in
Fig. 8.15). Although not shown, at
K
the lattice is still pinned. Since this temperature is greater than
the estimated Tm at this field (i.e. 9.5 K), it is likely that
in the expression for the melting line transition.
Figure 8.16 shows the measured field distributions in
``detwinned'' YBa2Cu3O6.60 (U2) as a function of magnetic
field at low temperatures. The asymmetries in the line shapes in the
top two panels are
similar to those for the twinned sample, showing the
features characteristic of a 3D vortex lattice. However,
at T where the line shape should exhibit the features
associated with the misaligned 2D-vortex lattices, the line shape
still shows a high-field tail. Note that the temperature in the bottom panel is
K, compared to
K
in the bottom panel of Figure 8.14--which makes it even more
surprising that the line shape is still asymmetric. Also, when the
detwinned sample is field cooled in a magnetic field of 3.4 T to
K, the measured field distribution still shows a
small high-field tail and the background signal can be field shifted.
The dimensional crossover observed in the twinned sample
should not arise from pinning by the twin boundaries,
since they extend the full depth of the sample and displace the 2D-vortex
lattices equivalently in all layers.
Since the two samples were not from the same batch,
the absence of the dimensional
crossover in the detwinned sample (U2) at fields comparable to those
in the twinned sample (U1), is likely related to differences in the
pointlike defects in the samples. In particular, these results suggest that
defects in the CuO2 layers are greater in the twinned sample.