Critical field studies on graphite intercalation compounds involve several experiments on a number of materials, are afflicted by many experimental difficulties, and are qualitatively described by a host of difficult microscopic theories. A critical review of the facts and speculations of the previous sections is in order to draw some conclusions.

From the experimental standpoint, there are several ways in
which this project could have been improved. Better cryogenic
instrumentation would have permitted the collection of more
accurate data at additional temperatures. Better structural
characterization as far as crystalline quality and in-plane
homogeneity would have taken some of the uncertainty out of
the interpretation of the data. The preparation of single
crystals of GIC superconductors would allow anisotropy
studies to be extended to a search for angular dependence of
H_{c2} in the layer planes.

The H_{c2}(*theta*) experiments, which were
affected more by the quality of the crystals than by
instrumental problems, raised some intriguing questions about
C_{4}KHg. The most interesting question is whether
C_{4}KHg has type I character for small angles
*theta*, and the answer seems to be yes, both from the
critical field and specific heat studies. The other
unexpected finding is that the angular dependence seems to be
better described by Tinkham's formula than by the anisotropic
Ginzburg-Landau model. The reason for the unusual shape of
H_{c2}(*theta*) may be understood only if
angle-dependent demagnetization effects can be accounted for
in a quantitative way, or perhaps if one of the microscopic
models of anisotropic superconductivity can be fit to the
data.

The most important finding of the angular dependence
experiments is that the magnitude of the linear specific heat
coefficient, *Gamma*, found by Alexander *et al.*
appears to be too high. One of the most longstanding
mysteries about GIC's has been why C_{4}KHg with N(0)
= 0.41 states / (eV*cdot*atom) has a lower T_{c}
than C_{8}KHg with N(0) = 0.058 states
/(eV*cdot*atom).[8] However, the *Gamma*
determined from these specific heat measurements predicts
such high values of H_{c}(t) that C_{4}KHg
would be type I for most angles *theta* (see Table
), in sharp contrast to the experimental observations.
The magnitude of H_{c}(t) determined from the
H_{c2}(*theta*) fits is about a factor of 1.8
lower than the magnitude of H_{c}(T) determined from
Alexander's specific heat data.

Since in weak-coupling superconductors H_{c}
*propto* *sqrt**Gamma*,[252] and *Gamma*
*propto* N(0), this finding implies that the
density-of-states measured for C_{4}KHg by Alexander
and colleagues may have been about a factor of 3.2 too high.
Then the corrected N(0) for C_{4}KHg = 0.13
states/(eV*cdot*atom). If the further assumption is made
that the specimen measured by Alexander *et al.* had a
T_{c} of 0.7 K (they measured only down to 0.8 K, and
saw no transition), then this N(0) is the phonon-dressed DOS,
which requires a further division by (1 +
*Lambda*_{ep}) = 1.38. (See Table
.) Then the final value of N(0) estimated from the
H_{c2}(*theta*) experiments is 0.094 states/(eV
*cdot*atom). This is still higher than the latest number
for the DOS of C_{8}KHg = 0.058 states
/(eV*cdot*atom) reported by Alexander,[8] but not higher than their
original estimate of N(0) for C_{8}KHg = 0.18 states
/(eV*cdot*atom).[7] Because C_{8}KHg is
more strongly type II than C_{4}KHg,[120] using the higher value of N(0)
would not imply type I behavior.

This extended discussion of the density-of-states issue is
not meant to produce hard numbers for C_{4}KHg or
C_{8}KHg. Rather the point of this inquiry is merely
to suggest that the ``mystery'' of why C_{4}KHg has a
lower T_{c} may not be so mysterious after all. A new
specific heat measurement on a C_{4}KHg specimen with
a known T_{c} would be valuable in clearing up these
issues.

The H_{c2}(T) measurements were more affected by
instrumental difficulties than the angular dependence
experiments. Nonetheless, two interesting anomalies were
found in the data: evidence for the temperature dependence of
*epsilon*, the anisotropy parameter; and enhanced
linearity of the critical fields. (There is no conflict
between these two observations because *epsilon* was
determined from the more accurate
H_{c2}(*theta*) experiments, and the amount of
change in d H_{c2}/dT implied by the variation of
*epsilon* is within the error bars of the
H_{c2}(T) measurements.) These two features of the
temperature dependence contradict the predictions of the AGL
model, which otherwise describe the data rather well.

The enhanced linearity of H_{c2}(T), temperature
variation of *epsilon*, and unexpected fits to Tinkham's
formula provide motivation for interpreting the data using
one of the many microscopic models available. There is ample
justification for doing so considering that the anomalies
seen in C_{4}KHg were observed on a larger scale in
the other ternaries[120] and
the binaries.[141,133] The most sensible models
to consider are those that fit the data of the closely
related transition metal dichalcogenide superconductors well.
Of these, the models dealing with Fermi surface and energy
gap anisotropy or multiband effects seem most promising. The
other enhancement mechanism that seems relevant to GIC's is
one developed by Carter and coworkers[34] which incorporates the effects
of microscopic inhomogeneity. Unfortunately a more complete
Fermi surface computation is needed for a ternary GIC before
these theories can directly tested.

To summarize the findings discussed in this chapter, let's
see what general statements can be made about ternary GIC
superconductors. First of all, they are weak-coupling
superconductors with *Lambda*_{ep} about 0.4,
which is typical for materials with T_{c} on the
order of 1 K.[165]
Orbital pairbreaking is the principal influence on the
critical fields of GIC's since they have *alpha*
10^{-3}, and so are not affected by Pauli-limiting
effects. Several GIC's ( C_{8}K, C_{8}Rb, and
C_{4}KHg) show signs of type I superconductivity for
a small angular region around *vec*H || ^c. Ternary GIC
superconductors have anisotropy ratios 1/*epsilon* in
the range from about 9 to 40, and they are fairly well
described by the anisotropic Ginzburg-Landau model, despite
some deviations. These deviations, which are outside the
scope of the AGL model, include a temperature-dependent
anisotropy and either extended linearity or positive
curvature of H_{c2}(T). All in all, the ternary GIC
superconductors show great similarity to the TMDC's, the
principal difference being that the TMDC's are more strongly
coupled, with higher T_{c}'s and larger critical
fields.

Figure
is offered as a final reminder of how much is unknown
about superconductivity in graphite intercalation compounds.
This figure shows the critical fields as a function of
temperature for the binary GIC's C_{8}K and
C_{6}K. C_{8}K is the simplest of all the
superconducting GIC's, and hence should be the first test of
any proposed model. As has been pointed out many times, the
very presence of superconductivity in C_{8}K is
something of a mystery since neither of its starting
constituents is superconducting alone.[234,4] C_{8}K has large positive
curvature of its critical fields, as the figure shows.
C_{6}K is a high-pressure phase of C_{8}K
which shows enhanced linearity rather than positive curvature
in H_{c2}(T).[13]
In this, and in its T_{c} = 1.5 K,[13] C_{6}K is strikingly
similar to C_{4}KHg. The challenge to critical-field
theorists here is quite clear: explain why the application of
pressure at the same time increases T_{c} tenfold and
suppresses positive curvature. Until basic questions about
the prototype GIC superconductor, C_{8}K, are
answered, one can hardly hope to have a complete
understanding of the much more complex ternary compounds.

**Figure:** Comparison of H_{c2}(T) in
C_{8}K and C_{6}K, one of its high-pressure
phases. a) Data on a T_{c} = 134 mK C_{8}K
sample taken by Koike and Tanuma.[141] Note the marked positive
curvature of the critical fields. H_{sc||} is a
supercooling field. b) Data on a T_{c} = 1.5 K sample
of C_{6}K from Ref. [13]. (*circ*), H_{c2, _|_
^c}; (*bigtriangleup*), H_{c2, || ^c}. Note
the enhanced linearity of the critical fields.

It would be a great omission to end a discussion of critical
fields in anisotropic superconductors without touching on the
central problem of the underlying cause of critical field
enhancement in anisotropic materials. Anyone who follows the
literature on anisotropic superconductors can plainly see
that nearly all layered materials have extended linearity or
positive curvature of H_{c2}(T), be they artificially
structured superlattices, high- T_{c}
superconductors, heavy-fermion or organic materials,
transition metal dichalcogenides and their intercalation
compounds, or even GIC's. The ubiquity of critical field
deviation above the WHH theory was previously noted by
Woollam, Somoano, and O'Connor.[267] The mass of accumulated
evidence forces one to look for a common origin of the
enhancement in these materials, even though they are each
described by their own special microscopic models: the
artificially structured superlattice by Takahashi and
Tachiki,[235] the
heavy-fermion materials by Delong *et al.*,[57], the organic compounds and
TMDCIC's by the KLB model,[131] the GIC's and TMDC and Chevrel
phases by the anisotropic gap/FS or multiband models,[33,81] and the high- T_{c}
superconductors by who-knows-what.[162]

Happily a common origin to the enhancement in all these
materials can be deduced. The key is the juxtaposition of two
superconducting components, either layers or bands, one of
which has a high intrinsic critical field, and the other of
which has a lower intrinsic critical field. Near
T_{c}, where the coherence lengths are large and
non-local effects are small, the two components are coupled,
and d H_{c2}/dT is determined by both jointly. That
is, an average sort of critical field results because the
system is forced to distribute field-induced vortices equally
over both components. At lower temperatures, the two
components are more weakly coupled, so that the
superconductor can concentrate the vortices in the
intrinsically low- H_{c2} component. This component
is now effectively normal even when H **<**
H_{c2}, so that the intrinsically high-
H_{c2} portion now determines the upper critical
field. Once the temperature is low enough that the high-
H_{c2} part decouples from the low- H_{c2}
part, d H_{c2}/dT will increase, causing positive
curvature.

As an example, consider S/N superlattices where one type of
layer (the N layer) has a low intrinsic H_{c2}. Near
T_{c} the vortices, which of characteristic size
*xi*(T), are too big to squeeze into the N layer, so
some average of the diffusivities of the N and S materials
determines the slope of H_{c2}. At lower temperatures
where vortices can fit into the N layers, those layers
effectively become normal at fields significantly below
H_{c2}. When they do, the lower-diffusivity S layers
determine d H_{c2}/dT, and the H_{c2}(T)
curve turns up. A detailed analysis of this situation has
been worked out by Takahashi and Tachiki, who predict a
first-order phase transition when the vortices fit into the N
layer.[236] This
transition may already have been observed experimentally on
Nb/Ta multilayers.[28,126]

The application of the idea of the decoupling of high- and
low- H_{c2} components to anisotropic gap/FS or
multiband models is less obvious, but similar. The
explanation has been given by Entel and Peter: ``For T ->
T_{c} and H_{c2}(T) -> 0 one should look
upon the condensation phenomenon as considering all electrons
in a small energy shell around the Fermi level even if local
regions on the Fermi surface contribute with a different
weight to T_{c}. But for T << T_{c},
the external field tends to suppress superconductivity in
those regions of the Fermi surface where the electrons are
more weakly coupled... the H_{c2}(0) enhancement is
merely determined by the strength of the off-diagonal
electron-phonon coupling constants...''[80] This interpretation is written
in general enough language that it can also be seen as
applying to a single anisotropic band. The similarity between
the decoupling between bands in a multiband superconductor
and layers in an artificially structured superlattice is
evident in comparing this description and that of Takahashi
and Tachiki.[236]

A unified theoretical picture for the critical fields of layered materials would be a great achievement. Such a unification appears inevitable as layer thicknesses in artificially structured materials get smaller and smaller, approaching the scale where Fermi surface anisotropy and multiband effects begin to take over. In fact, the possible observation of Fermi surface anisotropy effects in artificially structured superlattices has already been reported.[29] Undoubtedly new experiments will keep theorists busy for many years to come as they attempt to catch up with the experimental discoveries made possible by advances in fabrication.[209] It seems likely that the smaller layer thicknesses in artificial structures get, the more they will look like transition metal dichalcogenides and GIC's in their qualitative critical field behavior.

In this Chapter there has been mention several times of the
different critical fields of the gold low- T_{c} (
T_{c} 0.8K) and pink higher- T_{c} (
T_{c} 1.5 K) C_{4}KHg samples. Yet the
intrinsic cause of the differences between these types of
C_{4}KHg has not been discussed. The possible origin
of these differences and the insight that hydrogenation
experiments can give into this problem are the topic of the
next Chapter.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995