There are two-types of models which take anisotropy as a point of departure, rather than treating it as a small perturbation, as the pioneering Hohenberg and Werthamer model did.[111] One class consists of models that consider a single non-spherical Fermi surface and/or energy gap. This category includes work by Takanaka,[238] Pohl and Teichler,[195] Youngner and Klemm,[274] Butler,[33] and Prohammer and Schachinger.[201] The other class of theories study the implications of two different Fermi surface pieces that contribute to superconductivity. These models have been developed by Entel and Peter,[80,81] Decroux,[52], and Al-Jishi[4].

Theories in the first class, consisting of the anisotropic
gap/Fermi surface models, are essentially extensions of the
AGL model discussed previously. That is, these theories
contain the essential physics of the AGL model plus
modifications describing what are called non-local effects.
``Non-local'' is an adjective that refers to clean-limit
phenomena outside the the scope of Ginzburg-Landau theories
which are related to the finite spatial extent of the Cooper
pair.[252] That the more
complex models reduce to the AGL model in the local limit is
demonstrated by Hohenberg and Werthamer.[111] What Hohenberg and
Werthamer did was to rewrite the eigenvalue equation for the
energy gap (which is an intermediate step in the derivation
of the Helfand-Werthamer equation for H_{c2}) in
terms of an infinite sum. The sum has terms which are matrix
elements of even powers of a quantity proportional to
*vec*v *cdot* *vec**pi*, where
*vec*v is the Fermi velocity and *vec**pi* ==
*hbar*/i *vec**nabla* - e/c *vec*A is the
gauge-invariant momentum operator. Because of the
relationship between the vector potential *vec*A and the
magnetic field, it turns out that the component of
*vec*v parallel to *vec**pi* is orthogonal to
the applied field, in keeping with the expectation from the
GL theory that transport in a plane perpendicular to the
applied field determines H_{c2}. The important
quantity in this formalism is

where the integral is to be taken over the FS. Here
*vec*q is a point on the Fermi surface, N(*vec*q)
is the density of states at that point, and v_{_|_
^H} (*vec*q) is the component of
*vec*v_{F}(*vec*q) perpendicular to
*vec*H. Obviously the critical field anisotropy is
uniquely specified by one parameter *epsilon* only if
the FS is ellipsoidal. Otherwise knowledge of the entire FS
geometry is needed. Notice that v_{_|_ vecH}
(*vec*q) depends not only on the magnitude of
*vec*v_{F} at the point *vec*q, but also on
the angle between the angle between *vec*H and
*vec*v_{F}(*vec*q). This point is
illustrated by Fig. from Ref. [50].

**Figure:** Illustration of how v_{_|_
vecH} (*vec*q) changes as a function of
wavevector *vec*q for an ellipsoidal Fermi surface.
*vec*q is the coordinate of a point on the Fermi
surface. B is Dalrymple's anisotropy parameter, which is
equivalent to *epsilon* in the AGL model.[50]

Hohenberg and Werthamer went on to show that keeping only the
zeroth-order term in the infinite sum gives the isotropic
theory, while keeping terms up to the second order in the dot
product gives the AGL model, including Eqn.
for H_{c2}(*theta*).[53] What all subsequent models of
FS/gap anisotropy do is to include more terms in the sum over
powers of ( *vec*v *cdot* *vec**pi*).
Increasingly higher-order terms correspond both to higher
degrees of non-locality, and because of symmetry, also to
more complex FS/gap geometries ( *i.e.,* higher
angular momentum spherical harmonics). Hohenberg and
Werthamer included fourth-order contributions
perturbatively,[111]
which corresponds to the weak-anisotropy limit. The
weak-anisotropy limit is probably less appropriate for GIC's
than for superconductors like niobium or vanadium, which have
1/*epsilon* on the order of 2 or 3. The Hohenberg and
Werthamer model only modifies H_{c2}(t) near
T_{c}, and cannot produce the low-t enhancement
observed in GIC's.

Takanaka included terms up to order four in a
non-perturbative manner, but his theory is limited to t 1
because of an error.[274,114] Youngner and Klemm, who fit
their equations to NbSe_{2} data, found a way to sum
the series exactly up to infinite order.[274] They remark that their
model is basically equivalent to that of Pohl and Teichler,
who fit data on vanadium.[195] These models are also very
similar to the work of Butler,[33] who applied his theory to
niobium, except that Butler dropped all terms due to gap
anisotropy in his final equations because he thought were
quite small. Butler's equations have also been used by
Dalrymple and Prober[51] to fit data on
Nb_{1-x}Ta_{x}Se_{2}.

From examination of the h^{*} versus t curves that
appear in either Butler's paper[33] or Youngner and Klemm's
paper,[274] it is clear
that either one can produce both extended linearity using an
ellipsoidal Fermi surface model. This is encouraging because
the *pi*-bands of GIC's are generally believed to
produce an ellipsoidal FS piece,[67] but it is not the whole story. A
calculation using Butler's equations taken from Dalrymple's
thesis[50] is shown in
Fig. . Notice that
h^{*}(t) is enhanced only when Dalrymple's parameter
B (which is the same for an ellipsoidal Fermi surface as
*epsilon* from the AGL model) is less than one, and that
h^{*}(0) is actually suppressed from the isotropic
value when B is greater than one. The significance of this
conclusion is that for a simple ellipsoidal FS, the Butler
model extended linearity gives only for one field
orientation.[50]
Therefore one cannot describe critical field enhancement for
both field orientations using a simple ellipsoidal FS.

**Figure:** Enhanced linearity of
h^{*}(t) calculated from Butler's equations[33] using an ellipsoidal FS model.
Taken from Ref. [50].
Dalrymple's parameter B is equivalent to 1/*epsilon* in
the AGL model. The B = 1.0 curve is for a spherical Fermi
Surface, and so is equivalent to the WHH theory.

Because Dalrymple and Prober observed enhanced critical
fields for NbSe_{2} for both *vec*H || ^c and
*vec*H _|_ ^c,[51]
they tried fitting their data with more complex Fermi surface
geometries. When they combined an ellipsoid with a
cylindrical FS model calculated by Wexler and Woolley[263], they found excellent
quantitative agreement between theory and experiment for
*vec*H || ^c, and good qualitative agreement for
*vec*H _|_ ^c.[50]
These fits are shown in Fig. .
In order to get quantitative agreement for *vec*H _|_
^c, the authors had to multiply all the data through by a
factor of 2.1. Dalrymple and Prober explained that this
factor was necessary because of mean-free-path anisotropy,
which is not taken into account in the Butler model. The
mean-free path *l* comes in because of the dependence of
H_{c2} on the diffusivity D == 1/3 v_{F}
*l*. This type of mean-free-path anisotropy is quite
believable since much larger ratios have often been seen in
GIC's.[166] Essentially
Dalrymple and Prober find that the Butler model describes
H_{c2, || ^c} data without correction. This agreement
occurs since in-plane transport in NbSe_{2} is clean,
and Butler's model is for clean-limit superconductors. The
Butler model needs a correction to fit H_{c2, _|_ ^c}
data because transport along ^c is dirty.

**Figure:** Butler-model[33] fit of NbSe_{2}
H_{c2}(t) data. Figures taken from Dalrymple's
thesis.[50] a)
H_{c2, _|_} = H_{c2, || ^c}. An excellent fit
is obtained by using the Wexler-Woolley Fermi Surface
model[263] plus an
additional ellipsoid. b) H_{c2, ||} = H_{c2, _|_
^c}. The Wexler-Woolley-plus-ellipsoid model produces
the correct shape, but needs to be multiplied by an
additional factor of 2.1 to account for mean-free-path
anisotropy.

Youngner and Klemm's model of H_{c2}(T) seems to be
the same as Butler's, but with the added feature of gap
anisotropy.[274] In this
case, *vec*v(*vec*q) *cdot*
*vec**pi* may still depend on *vec*q. In
addition, though, the states between which the matrix element
of *vec*v *cdot* *vec**pi* is evaluated
may also themselves be non-spherical. Youngner's model
produces curves quite similar to those of Butler, including
extended linearity and positive curvature for some choices of
parameters. The recent theory of Prohammer and Schachinger[201] is nearly the same as
Youngner and Klemm's model, only these authors consider
electron-phonon coupling anisotropy rather than explicit gap
anisotropy.

As if these weren't already enough models, there are also
theories which consider the effect of multiple bands
contributing to superconductivity. The most extensive of
these was developed by Entel and Peter.[80] A two-band model fit by Entel
and Peter to data on
Cs_{0.1}WO_{2.9}F_{0.1}, a tungsten
fluoroxide bronze, is shown in Fig. .
Decroux and Fischer advocate the use of the Entel-Peter model
for fits to ternary molybdenum chalcogenide data.[53] The main parameters which
determine the critical field enhancement in this theory are
the interband and intraband scattering times. Al-Jishi[4] has proposed a theory quite
similar to Entel and Peter's, but his critical field
calculations are still in a preliminary phase. In GIC terms,
these models propose that intercalant and graphitic bands
both contribute to superconductivity. This idea was
originally suggested by Al-Jishi,[4] and it is supported by a large
amount of experimental evidence, as is described in Section
.

**Figure:** Two-band model fit to anomalous
H_{c2}(t) of
Cs_{0.1}WO_{2.9}F_{0.1} from Ref. [81]. The plot is of
h^{*} versus t. The curve labeled (4) is the
Helfand-Werthamer isotropic theory. The crosses, circles and
squares are experimental data for three different
crystallographic orientations (the orientations are not
specified). Curve (1) is the two-band model with no
interband-scattering, whereas (2) and (3) correspond to
increasing interband-scattering. The parameters of these fits
are too numerous to list here, but may be found in Ref. [81].

The profusion of models which predict an enhanced
H_{c2}(T) is quite confusing, especially since both
the multiband and anisotropic gap/FS theories seem to have
features which are very sensible for GIC's. Prohammer and
Schachinger say in their recent paper[201] that the Entel-Peter model
and the anisotropic gap/FS models are actually equivalent,
really amounting to different parameterizations of the same
phenomena. Considering the similarities of the curves in
Figs. and ,
this is not too surprising.

The question of whether these models will explain the
anomalies in GIC H_{c2}(T) data, ranging from the
extended linearity of C_{4}KHg to the positive
curvature of C_{8}RbHg, still has not been directly
addressed. Clearly all of the multiband and anisotropic
gap/FS theories are capable of producing curves of the right
shape, but this doesn't mean that they would give good fits
to the data using reasonable FS/gap parameters.
Unfortunately, no complete theoretical Fermi surface
computation for the ternary GIC's is available, and there is
only limited knowledge about the normal-state transport
properties. Holzwarth[112] has calculated the band
structure of C_{4}KHg, but published only a
qualitative sketch of the Fermi surface. This sketch is
reproduced in Fig. . If Holzwarth publishes
quantitative information about the FS of C_{4}KHg,
or, even better, FS information for C_{8}RbHg, then a
fit could be made to the critical field data. Obviously this
hypothetical fit would be the decisive test of the
applicability of these models.

**Figure:** Fermi surface computed for
C_{4}KHg by Holzwarth and colleagues.[112] The basic structure of the
Fermi surface is similar to that of NbSe_{2}[51] in that both have pieces of
nearly cylindrical symmetry at the corner of a hexagonal
Brillouin zone, and both have higher masses for transport
along k_{z} than in the layer planes. The hexagonal
solid line is the Brillouin zone; the roughly triangular
pieces drawn with a solid line at the corners of the BZ are
the graphitic *pi* bands. The pieces drawn with a dotted
line are due to mercury bands. The small circular zone-center
part is from Hg 6**s** holes; the hexagonal portion is
from Hg 6p*pi* electron carriers; and the trigonal
pieces at the zone corner are derived from H 6p*sigma*
bands.

Lamentably an attempt to invert the critical field data to
predict the geometry of the Fermi surface is not justified by
the critical field data which is available. In principle at
least, if H_{c2}(T) had been measured at a large
number of angles *theta*, this inversion could be
performed.

What conclusions can be drawn about the appropriateness of
the anisotropic gap/FS or multiband models for GIC's in the
absence of a fit? On the plus side, these models contain
features compatible with what is already known about GIC's:
GIC's have highly anisotropic Fermi surfaces; they are more
disordered along the c-axis than in-plane; they have multiple
bands present at the Fermi surface; and their
h^{*}(t) curves are much like those of the transition
metal dichalcogenides for whom most of the models were
intended. Therefore the tentative conclusion is that GIC
superconductivity is described by one of these theories,
although for positive confirmation, fits to the data are
still needed.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995