Another peculiarity of the C4KHg Hc2(theta) curves
is that they are well-fit by Tinkham's Hc2(theta) formula for
thin films:[250]
The Tinkham formula fit is decent (cal R = 0.83 at t = 0.29
for a Tc = 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 Hc2(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 Tc = 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- Tc 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 105 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 Hc2(theta) results from one sample to
another argue strongly against this interpretation.
Tinkham's formula has been used to fit Hc2(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 C4KHg is clearly well into the 3D
regime, where Eqn.
is supposed to be applicable. The 3D
nature of superconductivity in C4KHg 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 Hc2(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 Hc2(theta) data on
a Tc = 1.5 K C4KHg-GIC. bullet, data at t = 0.55.
circ, AGL fit with Hc2(0°) = 19 Oe, 1/epsilon = 15.5,
and a residual cal R = 0.84. diamond, TF fit with Hc2(0°) = 23 Oe, 1/epsilon = 13, Hc = 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 C4KHg 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 C4KHg : the gold lower- Tc
phase which is well-fit by Eqn.
would be due to bulk
superconductivity, whereas the pink higher- Tc 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 Hc2(theta) shape predicted by
Eqn. . Surface superconductivity cannot be responsible for
the Hc2(theta) behavior here since it can give only the
anisotropy ratio 1.69 = Hc3/ Hc2.[251]
There is one major problem with the idea of the pink ( Tc =
1.5 K) phase of C4KHg being present only as a surface film. That
problem is that if thin films are responsible for the angular dependence of
Hc2, 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 Hc2, _|_ ^c propto (1 - t)1/2, a form which is in definite conflict with the linear
behavior found for Hc2, _|_ ^c(t). (The temperature
dependence of Hc2 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 Hc2(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 vecH _|_ ^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 Hc2(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 Hc2(theta) data for the Tc = 1.5 K specimens. However, the
assumptions used in the derivation of the Tinkham formula[250]
imply a temperature dependence of Hc2 which is strongly in
conflict with experimental data. Therefore, solving the Hc2(theta) problems of the C4KHg data with Tinkham's formula
creates new problems for Hc2(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 Hc2(theta).