Another peculiarity of the C_{4}KHg
H_{c2}(*theta*) curves is that they are well-fit
by Tinkham's H_{c2}(*theta*) formula for thin
films:[250]

The Tinkham formula fit is decent (*cal* R = 0.83 at t =
0.29 for a T_{c} = 1.54 K sample) even without
allowing for type I behavior, and is quite good (*cal* R
= 0.26) if type I behavior is taken into account. A summary
of the residuals for fits to the H_{c2}(*theta*)
data is given in Table , and a direct comparison
between the two types of fits is shown in Figure
a). Figure b) shows the errors of
the two fits as a function of the angle *theta*. For the
T_{c} = 1.5 K GIC's the Tinkham formula fits have
residuals about half of those of the best AGL fits.
Consultation of standard tables on statistics[21] shows that (for 39 degrees
of freedom and a 3-parameter fit, the conditions of Figure
) there is about an 80% probability that the Tinkham
formula is a better description of the data than the AGL
model. The data on the lower- T_{c} samples are still
best fit by the plain two-parameter AGL theory, just as was
reported by Iye and Tanuma.[120]

The reason that a good fit to Eqn.
is unexpected is that its derivation requires the assumption
that the superconducting order parameter doesn't vary along
the direction perpendicular to the specimen's surface.[250] This assumption seems
reasonable for a film which has a thickness less than a
superconducting coherence length, but is hard to believe when
the sample is 10^{5} coherence lengths in thickness.
Only if the sample were so disordered along the c-axis that
its structural coherence length was on the order of the
superconducting coherence length could one envision that the
Tinkham formula should apply. However, the metallic nature of
the c-axis conductivity[85] and the reproducibility of
the H_{c2}(*theta*) results from one sample to
another argue strongly against this interpretation.

Tinkham's formula has been used to fit
H_{c2}(*theta*) curves in the artificially
structured superlattice superconductors, such as Nb/Cu[42] and Nb/Ta[29]. However in these systems
Eqn. fits only below the
3D-2D crossover point (see Chapter
), and C_{4}KHg is clearly well into the 3D regime,
where Eqn. is supposed to be
applicable. The 3D nature of superconductivity in
C_{4}KHg is unquestionable because the
Klemm-Luther-Beasley **r**-parameter is about 2000 at 0 K,
while 1 is the critical value for the dimensionality
crossover.[131] (The
question of dimensionality crossover as it relates to GIC's
is discussed further in Sections
and
.)

**Table:** Residuals for fits to
H_{c2}(*theta*) using the AGL formula and the
Tinkham formula, both with and without type I behavior. The
residual index *cal* R is defined in Eqn. ,
the AGL formula is Eqn. , and the Tinkham
formula is Eqn. . Eqn.
shows how each of these formulae was modified to account for
possible type I superconductivity.

**Figure:** Comparison of the Tinkham formula
and AGL theory fits to H_{c2}(*theta*) data on a
T_{c} = 1.5 K C_{4}KHg-GIC. *bullet*,
data at t = 0.55. *circ*, AGL fit with
H_{c2}(0°) = 19 Oe, 1/*epsilon* = 15.5, and
a residual *cal* R = 0.84. *diamond*, TF fit with
H_{c2}(0°) = 23 Oe, 1/*epsilon* = 13,
H_{c} = 41 Oe, and *cal* R = 0.47. Below, a plot
of the errors of each fit versus *theta*. The same
symbols are used.

Despite the statistically significant improvement that
Tinkham's formula gives over the AGL Eqn. ,
it is quite hard to justify the use of Tinkham's formula
theoretically. One obvious possibility is that the
superconductivity measured in C_{4}KHg is not a bulk
phenomenon, but is merely due to a thin layer on the surface.
The idea is not that there is surface-nucleated
superconductivity at the edge of a homogeneous bulk
superconductor, but that there could be a second
crystallographic phase stable only near the surface. This
explanation is appealing because it would help explain the
mystery as to the difference between the gold and pink phases
of C_{4}KHg : the gold lower- T_{c} phase
which is well-fit by Eqn. would be due to bulk
superconductivity, whereas the pink higher- T_{c}
phase would be present only on the surface. A surface phase
present only in a thin layer could easily mimic
two-dimensional behavior, and would be expected to fit
Tinkham's formula.

Superconductivity in a surface layer has already been shown
to give good agreement with Tinkham's formula. In lead films
doped with thallium, the ``thin film'' was a
surface-nucleated layer about a coherence length thick.[251] Nonetheless the
``film'' had critical fields which agreed quite well with the
H_{c2}(*theta*) shape predicted by Eqn. .
Surface superconductivity cannot be responsible for the
H_{c2}(*theta*) behavior here since it can give
only the anisotropy ratio 1.69 = H_{c3}/
H_{c2}.[251]

There is one major problem with the idea of the pink (
T_{c} = 1.5 K) phase of C_{4}KHg being
present only as a surface film. That problem is that if thin
films are responsible for the angular dependence of
H_{c2}, they should give[252]

where d is the thickness of the film. As discussed in Section
, thin films have a parallel critical field with a
temperature dependence of the form H_{c2, _|_ ^c}
*propto* (1 - t)^{1/2}, a form which is in
definite conflict with the linear behavior found for
H_{c2, _|_ ^c}(t). (The temperature dependence of
H_{c2} at constant angle is discussed in detail in
the Section .)

For a given anisotropy, the primary difference between
Tinkham's formula and Eqn. is that the Tinkham's
formula curve has upward positive curvature for all angles,
whereas the AGL angular dependence has an inflection point in
the wings of the peak, and has negative curvature at
*theta* = 90°. The challenge in justifying the
application of Tinkham's formula to GIC's is to think of a
factor that could cause H_{c2}(*theta*) to rise
more steeply than the AGL dependence. One aspect of the
problem that is completely overlooked in the derivation of
Eqn. (and in the derivation
of Tinkham's formula, for that matter) is the microscopic
physics of the flux-line lattice (FLL). In an isotropic
superconductor, in the absence of defects, the axis of
symmetry of a vortex must lie along the applied field. If the
symmetry axis of the vortex were not along the applied field,
the screening currents in the vortex would have to be larger,
which would cost kinetic energy. Tilley,[244] one of the originators of the
AGL model, calculated the properties of the flux-line lattice
(FLL) in anisotropic superconductors. He found that the
energy of the FLL is lowest when the applied field is along a
crystallographic symmetry direction. When the applied field
is at an arbitrary angle *theta* with respect to the
symmetry axes, the vortices pay a potential energy price for
their misorientation with respect to the crystalline axes.[244] Perhaps when the applied
field is oriented only slightly off a crystallographic
symmetry direction (such as *vec*H _|_ ^c), the
flux-line lattice might actually minimize its total energy
(kinetic energy from screening currents plus potential energy
of misorientation) by orienting the vortices' symmetry axes
along the crystallographic symmetry direction rather than
along the applied field. Quantitative calculations by
Kogan[138] and Kogan and
Clem[139] suggest that
this rotation of the flux-line lattice may be a common
feature of anisotropic superconductors. It is proposed that
rotation of the FLL to the crystallographic symmetry
direction for *theta* near 90° could help explain
why the experimental data rise more sharply than the AGL
model in the wings of the H_{c2}(*theta*) peak.
Unfortunately, coming up with a test of this hypothesis is
not easy. Magnetic torque measurements might be able to give
some information.[100]
Kogan and Clem suggest neutron scattering and nuclear
magnetic resonance tests.[139]

In summary, Tinkham's formula provides the best fit to the
H_{c2}(*theta*) data for the T_{c} = 1.5
K specimens. However, the assumptions used in the derivation
of the Tinkham formula[250] imply a temperature
dependence of H_{c2} which is strongly in conflict
with experimental data. Therefore, solving the
H_{c2}(*theta*) problems of the C_{4}KHg
data with Tinkham's formula creates new problems for
H_{c2}(t). Figure
b) shows that the Tinkham's formula fit seems to be an
improvement on the AGL theory principally in the wings, where
the angular dependence of the demagnetization may play a
role. Therefore an unsatisfying but reasonable conclusion is
that the agreement with Tinkham's formula is fortuitous,
implying that the formula somehow mimics the combination of
anisotropy, tilt, type I behavior, demagnetization effects
and mosaic spread that were actually present in the
experiments. Alternatively, the microscopic details of the
flux-line lattice may be having an effect on
H_{c2}(*theta*).

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995