The study of high-T_{c} superconductors is still very
new, so the critical field properties are still being sorted
out. Even though the number of papers published on the copper
oxides must already be several times the number published on
the TMDC's and the synthetic superlattices, there are several
difficulties standing in the way of a consistent
understanding of the properties of these materials. The
principal experimental difficulty is the (currently
unobtainable) high magnetic fields needed to measure
H_{c2} in an extensive temperature range, a problem
which is exacerbated by the quite broad transitions seen in a
magnetic field. Even pulsed magnetic field experiments do not
allow measurement of H_{c2, _|_ ^c} below about t =
0.85,[189,211] which means that any
conclusions drawn must be considered tentative because of the
limited magnetic field range of the data.

The other major difficulty with the high-temperature critical
field measurements so far is the uncertain theoretical
picture of these new materials. Until a theory of high-
T_{c} superconductivity earns widespread acceptance,
experimentalists must rely on the formulae of low-
T_{c} superconductivity, such as the WHHM theory and
the anisotropic Ginzburg-Landau theory (see Section 4.5).
Interpretation of the data in terms of these models may well
be misleading, but they are the best anyone can do under the
present circumstances.

Even with all these warnings in mind, the data that have been
published so far on the high- T_{c} materials are
intriguing to those familiar with previous research on
anisotropic superconductors. Some of the parameters from
recent experiments on the high- T_{c} superconductors
are gathered in Table 2.4. The angular dependence of
H_{c2} measurements are the most straighforward to
interpret since they are the least affected by magnetic field
limitations. H_{c2}(*theta*) data taken so far
shows that the 90 K rare-earth-barium-copper-oxide compounds
are 3D superconductors, well fit by Eqn. 4.3.
H_{c2}(*theta*) results on
HoBa_{2}Cu_{3}O_{x} from the work of
Iye *et al.* are shown in 2.8.

**Figure 2.8:** H_{c2}(*theta*)
for a high- T_{c} superconductor. From Ref. [118]. Each set of symbols
corresponds to a different definition of H_{c2}. For
example, the curve labeled 0.7 R _{N} was obtained by
plotting versus *theta* the values of H that satisfy
R(H,*theta*) = 0.7 R_{N}. Each curve is also
labeled in parentheses with the magnitude of the anisotropy
parameter 1/*epsilon* that was used for the fit to Eqn.
4.3.

A series of H_{c2}(*theta*) curves taken at
different temperatures indicates that the anisotropy
parameter *epsilon* is temperature-dependendent in these
materials.[118] A
temperature-dependent *epsilon* has been widely observed
in layered materials such as Nb_{(1 -
x)}Ta_{x}Se_{2},[51] C_{8}K[141], C_{8}KHg,[35] and Nb/Cu superlattices[42], where it has been associated
with positive curvature of H_{c2}(T) . As noted
previously, positive curvature of the critical field parallel
to the layer planes (_|_ to the c-axis) may be associated
with a dimensionality crossover, but the positive curvature
often observed in H_{c2} perpendicular to the layer
planes must have a different cause. In the high-
T_{c} superconductors, positive curvature has been
observed for both field orientations by some research
groups,[119,118,173,211] while a linear
temperature dependence has been reported for both field
orientations by others.[188,189,255,257] Worthington *et
al.*, on the other hand, see no sign of positive
curvature, but instead a change in the linear slope of
H_{c2}.[268]
These disagreements about the positive curvature and
temperature-dependent *epsilon* are reminiscent of a
similar controversy about NbSe_{2} in the '70's.[87,181] Whether the disparaties among
the various research groups are due to differences in sample
quality or slightly different methods of data analysis
remains to be seen.

**Table 2.4:** Selected properties of some of
the high- T_{c} superconductors. *dagger*
indicates a single-crystal measurement. The coherence length
quoted for ceramic sample is an average one. NR means that
the parameter was not reported in the cited reference.

Clearly a lot more work has to be done before the
H_{c2}(T) data in the copper-oxides can be
understood. One of the major questions still to be answered
is whether the superconductivity in these materials becomes
two-dimensional at low temperatures. The good fit of Eqn. 4.3
(rather than Tinkham's formula) to the data suggests that the
copper-oxides are 3D superconductors at least in the
temperature range accessible with currently obtainable
fields. The coherence lengths which have been measured so far
support the identification of 3D coupling.[268] However, a decoupling of
the copper-oxygen planes is suggested by the fact that the
presence of magnetic rare earth ions between the planes does
not seem to suppress the transition temperature.[239,188] Also, Table 2.4 shows that
the coherence length || to the c-axis is expected to be on
the order of the c-axis lattice constant at low-temperature,
which suggests that a coupling-dimensionality change might
occur. For example, for
HoBa_{2}Cu_{3}O_{(7-y)} the c-axis
lattice parameter is 11.67 Å[239] and *xi*_{||
^c} is about 6.5 Å. Plugging these numbers into
Eqn. 2.4 gives r = 1.58, slightly less than the critical
value of 1.7. Obviously estimates of this type are not to be
taken too seriously considering the large extrapolation.

Another interesting question is whether the copper-oxides'
transition temperatures follow the predictions of the
proximity effect. If the copper-oxygen planes are truly the
superconducting layers, then putting several of them together
without separation by an insulating layer should increase
T_{c}, according to proximity-effect models. Exactly
this type of enhancement has been seen in the
thallium-barium-calcium-copper-oxygen superconductors, where
T_{c} increases as a function of the number of
copper-oxygen planes that are adjacent.[197] Of course this T_{c}
behavior is not proof of proximity-effect behavior since
another mechanism could easily be responsible if the pairing
in these materials is non-conventional.

The picture from the currently available papers on high-
T_{c} superconductivity is obviously quite
preliminary. So far they appear most like the transition
metal dichalcogenides in their critical field properties:
they have an anisotropy of about 5,
H_{c2}(*theta*) well-fit by Eqn. 4.3, and
possible positive curvature of H_{c2}(T) and
temperature-dependent anisotropy. Even if the
superconductivity of the high- T_{c} materials turns
out to be mediated by an exotic interaction, the comparison
with the superconducting TMDC's and GIC's will still be
instructive.

Send mail to alchaiken@gmail.com (Alison Chaiken) if you would like a hardcopy.