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 Hc2 in the layer planes.
The Hc2(theta) experiments, which were affected more by the quality of the crystals than by instrumental problems, raised some intriguing questions about C4KHg. The most interesting question is whether C4KHg 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 Hc2(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 C4KHg with N(0)
= 0.41 states / (eVcdotatom) has a lower Tc
than C8KHg with N(0) = 0.058 states
/(eVcdotatom).[8] However, the Gamma
determined from these specific heat measurements predicts
such high values of Hc(t) that C4KHg
would be type I for most angles theta (see Table 
), in sharp contrast to the experimental observations.
The magnitude of Hc(t) determined from the
Hc2(theta) fits is about a factor of 1.8
lower than the magnitude of Hc(T) determined from
Alexander's specific heat data.
Since in weak-coupling superconductors Hc
propto sqrtGamma,[252] and Gamma
propto N(0), this finding implies that the
density-of-states measured for C4KHg by Alexander
and colleagues may have been about a factor of 3.2 too high.
Then the corrected N(0) for C4KHg = 0.13
states/(eVcdotatom). If the further assumption is made
that the specimen measured by Alexander et al. had a
Tc 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 +
Lambdaep) = 1.38. (See Table
.) Then the final value of N(0) estimated from the
Hc2(theta) experiments is 0.094 states/(eV
cdotatom). This is still higher than the latest number
for the DOS of C8KHg = 0.058 states
/(eVcdotatom) reported by Alexander,[8] but not higher than their
original estimate of N(0) for C8KHg = 0.18 states
/(eVcdotatom).[7] Because C8KHg is
more strongly type II than C4KHg,[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 C4KHg or C8KHg. Rather the point of this inquiry is merely to suggest that the ``mystery'' of why C4KHg has a lower Tc may not be so mysterious after all. A new specific heat measurement on a C4KHg specimen with a known Tc would be valuable in clearing up these issues.
The Hc2(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 Hc2(theta) experiments, and the amount of change in d Hc2/dT implied by the variation of epsilon is within the error bars of the Hc2(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 Hc2(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 C4KHg 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 Lambdaep about 0.4, which is typical for materials with Tc 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 ( C8K, C8Rb, and C4KHg) show signs of type I superconductivity for a small angular region around vecH || ^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 Hc2(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 Tc'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 C8K and
C6K. C8K 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 C8K is
something of a mystery since neither of its starting
constituents is superconducting alone.[234,4] C8K has large positive
curvature of its critical fields, as the figure shows.
C6K is a high-pressure phase of C8K
which shows enhanced linearity rather than positive curvature
in Hc2(T).[13]
In this, and in its Tc = 1.5 K,[13] C6K is strikingly
similar to C4KHg. The challenge to critical-field
theorists here is quite clear: explain why the application of
pressure at the same time increases Tc tenfold and
suppresses positive curvature. Until basic questions about
the prototype GIC superconductor, C8K, are
answered, one can hardly hope to have a complete
understanding of the much more complex ternary compounds.
Figure: Comparison of Hc2(T) in
C8K and C6K, one of its high-pressure
phases. a) Data on a Tc = 134 mK C8K
sample taken by Koike and Tanuma.[141] Note the marked positive
curvature of the critical fields. Hsc|| is a
supercooling field. b) Data on a Tc = 1.5 K sample
of C6K from Ref. [13]. (circ), Hc2, _|_
^c; (bigtriangleup), Hc2, || ^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 Hc2(T), be they artificially structured superlattices, high- Tc 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- Tc 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 Tc, where the coherence lengths are large and non-local effects are small, the two components are coupled, and d Hc2/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- Hc2 component. This component is now effectively normal even when H < Hc2, so that the intrinsically high- Hc2 portion now determines the upper critical field. Once the temperature is low enough that the high- Hc2 part decouples from the low- Hc2 part, d Hc2/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 Hc2. Near Tc 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 Hc2. At lower temperatures where vortices can fit into the N layers, those layers effectively become normal at fields significantly below Hc2. When they do, the lower-diffusivity S layers determine d Hc2/dT, and the Hc2(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- Hc2 components to anisotropic gap/FS or multiband models is less obvious, but similar. The explanation has been given by Entel and Peter: ``For T -> Tc and Hc2(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 Tc. But for T << Tc, the external field tends to suppress superconductivity in those regions of the Fermi surface where the electrons are more weakly coupled... the Hc2(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- Tc ( Tc 0.8K) and pink higher- Tc ( Tc 1.5 K) C4KHg samples. Yet the intrinsic cause of the differences between these types of C4KHg 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.