This section is not intended to be a complete review of
H_{c2}(T) theories. Instead, the following is a
discussion of a few specific models which predict enhanced
critical fields and the prospects for the application of
these models to GIC's. A recent complete overview of
H_{c2}(T) theories has been written by Decroux and
Fischer.[53]

Before reviewing the models which help to explain the
deviations from the anisotropic Ginzburg-Landau theory, it
makes sense to review the basic AGL model itself. The
anisotropic Ginzburg-Landau model, like the Ginzburg-Landau
theory[252] from which it
is descended, is based on an expansion of the free-energy
difference between the superconducting and normal phases into
a series of powers of the order parameter. For
superconductivity, the order parameter is variously taken as
the energy gap (in theories based on the Gor'kov equations[94,93]), the density of superfluid
particles (in the spirit of the two-fluid model), or the
Ginzburg-Landau wave function *psi*. In the simplest
case of the isotropic GL theory, the free-energy difference
of the superconducting and normal states is written:

Here *alpha* is the kinetic energy of the Cooper pair,
and *vec*A is the vector potential. The GL differential
equations are derived from the free-energy variationally.
Their solution for the zero-field case shows that |
*alpha* | = *hbar*^{2} / 2m^{*}
*xi*^{2}, where *xi* is the superconducting
coherence length discussed in Section .
Using the relation H_{c2} = ø_{0} /
2*pi* *xi* ^{2} and rearranging factors
shows that *alpha* is equal to the cyclotron energy of
an electron at H = H_{c2}. This equality points to
the connection between vortices in type II superconductors
and the Landau-level problem which is mentioned in Tinkham's
book.[252, Section 4-8,]

The only elaboration necessary to go from the isotropic to
the anisotropic case is to let the mass m^{*} become
a tensor rather than a scalar.[244,127] All the rest of the results of
the AGL model follow directly from this replacement. The
spirit of the AGL model is to let all the orientation
dependence be lumped into the masses, and to ignore any
variation with angle of other microscopic parameters
(*e.g.,* mean free path and phonon dispersion).
Therefore, all anisotropy effects are described in this model
by one temperature-independent parameter, *epsilon*.
This parameter was introduced by Morris, Coleman, and
Bhandari[175] who defined
*epsilon* == *sqrt*m/M, where m is the in-plane
mass and M is the c-axis mass. With the assumption of
in-plane isotropy, further calculation shows that H_{c2,
|| ^c}/H_{c2, _|_ ^c} = *xi*_{_|_
^c}/*xi*_{|| ^c}. Because *epsilon* is
the sum of all the contributions to the anisotropy of a
layered superconductor, it should not necessarily be thought
of as the ratio of band masses. Similarly, the fact that the
flux quantum in the AGL model is ellipsoidal does not imply
the presence of an ellipsoidal Fermi surface.

The simplicity of the AGL model is both its greatest virtue
and its greatest weakness. On one hand, the small number of
parameters in the model allow calculation of quite complex
quantities, such as the angular dependence of the lower
critical field, H_{c1}.[132] No sane person would seriously
contemplate the calculation of H_{c1}(*theta*)
in the context of a first-principles model of anisotropic
superconductivity, where closed-form final expressions are
rare. On the other hand, the AGL model is so simple that it
is inadequate to describe all the details of
superconductivity in GIC's, despite the fact that it gives a
satisfactory picture of the main features. Both the poor fits
to H_{c2}(*theta*) (see Section )
and the temperature dependence of *epsilon* (see Section
) cannot be explained in the context of the AGL model.
Furthermore, the enhanced linearity of H_{c2}(T)
cannot be explained within the WHH theory, which reduces to
the AGL model for T T_{c}.[94,84] Considering the many aspects
of anisotropy in GIC's[67], it
is not surprising that the one parameter *epsilon* is
insufficient to summarize all their manifestations. All in
all, the anisotropic Ginzburg-Landau model is successful
within the confines of its applicability, and useful as far
as the physical insight it provides, but it is not the last
word on the subject by any means.

Before going into which models are the most appropriate for a
detailed application to C_{4}KHg, it is useful to
quickly discuss those possibilities which produce similar
effects in the data, but are not applicable here. Most of
these models are extensions of the WHH theory, but one, the
Klemm-Luther-Beasley (KLB) dimensionality-crossover model, is
based more directly on the AGL model.

- Dimensional Crossover Models
- Critical Field Enhancement in Isotropic Superconductors
- Multiband and Anisotropic Gap Models of Superconductivity

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995