The direct ancestor of the KLB papers, and one of the primary contributions to the AGL model, is the Josephson-coupling theory of Lawrence and Doniach[155]. Lawrence and Doniach started with the idea of a superconductor-insulator (S/I) superlattice whose layers are coupled by Josephson tunnelling of quasiparticles. They modified the GL free-energy expression accordingly, and found an effective mass along the c-direction in terms of the tunnelling parameters. From this, following the work of Kats[127], they showed that the critical field anisotropy was equal to the square root of the ratio of the effective masses.

Klemm, Luther and Beasley[131] studied the GL equations with
Josephson-coupling further, and found that they predicted a
divergence of H_{c2, _|_ ^c}(T) at a temperature
T^{*} given by *xi*_{||
^c}(T^{*}) = s/*sqrt*2. In this equation, s
is the interlayer spacing corresponding to I_{c} in
GIC's. KLB then did a first-principles calculation of
H_{c2}(*theta*, T) and discovered that
T^{*} is actually the temperature at which a
dimensionality crossover occurs. Dimensionality crossover in
the context of layered superconductors means a change in
character from bulklike behavior to film-like behavior. This
change is caused by a decoupling of the superconducting
layers from one another when *xi*_{||^c} becomes
less than s. The symptoms of 2D behavior are that the
H_{c2}(*theta*) curves are fit with Tinkham's
formula rather than the AGL angular dependence formula, and
that H_{c2}(T) goes as (1 - t)^{1/2} rather
than the (1 - t) shape found near T_{c} in 3D
superconductors. This type of dimensionality crossover has
been experimentally observed both in TMDCIC's[200] and artificially structured
superlattices (which are discussed in Chapter
).[208] As an
explanation for anomalies in superconducting GIC's,
dimensionality crossover has a lot of appeal, since so many
of the normal state properties of GIC's are
quasi-two-dimensional. For example, the anisotropy of the
conductivity in C_{4}KHg is
*sigma*_{a}/*sigma*_{c} 300.[85]

**Figure:** Theoretical demonstration of
dimensional crossover in Josephson-coupled superlattices from
the work of Klemm, Luther and Beasley.[131]. r is the parameter which
characterizes the dimensionality of coupling. *alpha*,
*tau*_{SO}, and H _{P} == 4
k_{B} T_{c}/ *pi* *mu* are
parameters which characterize the degree of Pauli-limiting
(Pauli-limiting is discussed in Section ).
The inset shows a plot of T^{*}/ T_{c} (where
T^{*} is the dimensionality crossover temperature)
versus r.

According to KLB, a layered superconductor will have the
possibility of a coupling-dimensionality change when
T^{*} is greater than zero, which occurs when their
parameter **r** is close to 1. The exact condition
is

Calculated **r** values for a number of layered
superconductors are shown in Tables and . The r's for
the GIC superconductors are shown in Fig. .
As can easily be seen, many of the transition metal
dichalcogenide superconductors and their intercalation
compounds are expected to show the dimensionality crossover
effect, and indeed many do, as was discussed in Section .[46,200] On the other hand, the known
GIC superconductors are more than an order of magnitude away
from the critical value of r. The KLB paper found no
anomalies for **r** >= 10,[131] as Fig.
shows. Therefore, dimensionality effects can safely be ruled
out as having any impact on the critical fields of the known
GIC superconductors. This is particularly interesting in
light of the fact that many GIC's have anisotropies
1/*epsilon* comparable to those of the TMDCIC's. For
example, both C_{8}KHg and
TaS_{1.6}Se_{0.4} intercalated with collidine
have anisotropies of about 30, yet the TMDCIC is expected to
show a dimensionality crossover, and C_{8}KHg is
strongly three-dimensional even at 0 K.

**Table:** Values of the KLB parameter **r**
for the GIC superconductors. means that the value of **r** was
calculated from parameters in the cited references.

At this juncture it is probably wise to discuss the proximity-coupled superlattice models of superconductivity[22,235], which have some similarities to the KLB model. These models are also capable of quantitatively explaining dimensional crossover, but in slightly different systems. Proximity-coupled superlattices are typically superconductor-normal-metal (SN) multilayers, while the Josephson-coupled superlattices described by the KLB theory are typically superconductor-insulator (S/I) multilayers.

The motivation for making this distinction was originally explained by Werthamer, and separately by Saint-James and deGennes.[261,54] At an S/I interface (the insulator may be vacuum), the pairs are confined in the superconductor so that they are almost unaffected by the presence of the boundary. In this case it is appropriate to ignore any variation of the order parameter within the superconducting layers, and to treat the interlayer coupling as due to Josephson tunnelling.[155] On the other hand, at an S/N interface, the pairs can easily diffuse into the normal metal, where they will be destroyed.[54] The destruction of pairs at an S/N contact results in the suppression of the energy gap at the interface, and is one aspect of the proximity effect. Another aspect is that the superconductor can to some extent induce superconductivity in a thin layer of the normal metal.[49] The proximity-coupled theories of Takahashi and Tachiki[235] and Biagi, Kogan, and Clem[22] (BKC) were developed to treat the S/N case.

In these models, the authors parameterize the S and N layers
separately, unlike the KLB theory, where the properties of
the insulating layers are ignored.[131] Takahashi and Tachiki[235] assign different
densities of states N(0), diffusivities D, and BCS
interaction energies V to the S and N layers. BKC, on the
other hand, attribute to each type of layer a mean-free-path
*l*, superconducting transition temperature
T_{c}, and Fermi velocity v_{F}.[22] Because v _{F} and
*l* determine D, while N(0) and V determine (with
*Omega*_{D}) T_{c}, it seems that these
parameterizations are more or less equivalent.

The full theories both predict positive curvature of
H_{c2}(T). Takahashi and Tachiki found signs of a
change in the coupling dimensionality for H_{c2, _|_
^c}(T).[235]
Qualitatively these results are quite similar to those of
KLB. The exact criteria for dimensional crossover are
different, though: the authors say that the necessary
condition for decoupling of the layers (at T **>** 0)
is s/ *xi*_{|| ^c}(0) >= 0.4. This condition
is equivalent, in terms of KLB's r-parameter, to **r**
<= 31.8. Thus the proximity-coupled multilayers decouple
at longer coherence lengths (*xi*_{|| ^c} =
2.5s) than the Josephson-coupled ones (*xi*_{||
^c} = 1.7s). Nonetheless, in the proximity-coupled
superlattice models, no dimensionality change is anticipated
for superconducting GIC's, which all exceed the limit in
**r** by at least a factor of 2. (The possibility of
dimensionality crossover in higher-stage GIC's is discussed
in Section .) The source of this
accelerated decoupling is not discussed in Ref. [235], but one can speculate
that it comes from the suppression of the gap near the S/N
interface by the proximity effect.

What is even more intriguing for those interested in GIC's is
the enhanced critical field found for *vec*H || ^c.
BKC[22] studied only this
case, and found that ``the proximity effect alone can produce
the positive curvature [PC] in H_{c2}(T) ... [despite
the fact that] no provisions were made for anisotropy or any
other effects commonly thought to produce PC in
H_{c2}(T).'' These remarks are a little bit deceptive
in that anisotropy has been implicitly introduced into the
BKC model through the layer thickness parameters; that is,
thin films are known to be anisotropic, and the individual
layers have been chosen to be thin films. Nonetheless, it is
impressive that positive curvature has been found in a
superlattice formalism which is not linked to a
dimensionality crossover. Takahashi and Tachiki also found PC
for *vec*H _|_ ^c.[235] These findings would seem
to have important implications for GIC's, for whom a possible
explanation of PC using dimensionality crossover has already
been eliminated.

Unfortunately there are several problems with applying the
proximity-coupled superlattice models to GIC's. One objection
is that assigning different mean-free-paths, diffusivities,
T_{c}'s and BCS interaction energies to the S and N
layers seems a bit questionable when the layers are only a
few atomic layers thick. These models were intended to be
applied to S/N superlattices with layers on the order of a
few-hundred Å thick, where each of the constituents
individually has transport properties which are little
modified from the bulk. For layers on the order of 10 Å
thick, the transport properties are so modified from the
pristine materials that use of the bulk parameters would be
highly erroneous. The basic assumptions of the
proximity-effect models are violated in GIC's, where the
interaction between the host and intercalant is not a weak
perturbation. Perhaps in the limit of very high-stage
superconducting GIC's (if they exist!) agreement with the
proximity-coupled description might be found.

A second reason for discomfort with the KBC and Takahashi and
Tachiki models is the idea that superconducting GIC's can be
described as S/N superlattices. While Iye and Tanuma felt
that superconductivity in the ternary GIC's was probably due
to superconductivity in the intercalant layers,[120] the idea that one component is
superconducting and the other not appears somewhat shaky when
*xi*_{|| ^c} 20 I_{c}, as in
C_{4}KHg. Also, as Al-Jishi pointed out, there is
good reason to expect that the graphitic *pi*-bands
contribute to superconductivity in GIC's, as otherwise their
large critical field anisotropy is difficult to explain.[4]

All these discussions are actually somewhat academic since
the idea that GIC's are S/N superlattices turns out not to be
self-consistent. The reason is that all proximity-effect
theories[49,235] predict that the
T_{c}'s of S/N multilayers must always be less than
the bulk transition temperature of the superconducting
component. In fact, these models predict that, for small
layer thicknesses, T_{c} must decline monotonically
with increasing thickness of the normal layer, all other
parameters being held equal. The monotonic decrease of
T_{c} with increasing normal metal thickness is
illustrated in Fig. for Pb/Cu bilayers.[261] Apparently no studies
of T_{c} versus normal metal thickness have been
performed in metallic superlattices; the thicknesses of both
the S and N components are usually varied together. (See
Figure .) However, the
depression of T_{c} with increasing normal metal
thickness is also expected in the superlattices. If one
believes that only the intercalant layer is superconducting
in GIC's, and thinks of the carbon (graphene) layer as a
normal metal, then one would expect a decreasing
T_{c} with increasing stage. Yet, as is well-known,
T_{c} is higher in stage II than stage I for both
KHg- and RbHg-GIC's,[116] in
contravention to the expectation of the model.

**Figure:** T_{c} versus normal-layer
thickness for S/N bilayers. Figure taken from Ref. [261] Here D _{N} and D
_{S} are the thicknesses of the normal and
superconducting layers, and T _{cS} is the bulk
T_{c} of the superconducting component. Approximately
T/ T_{cS} = (1 - t(D_{N} ->
*infty*))(1 - *exp*-2D_{N}/*xi*),
where *xi* is the dirty-limit Pippard coherence
length.

The conclusion is that, while the proximity-coupled models have been extremely successful in fitting data on artificially structured superlattices (for example, those of Nb/Ta[29]), they do not seem to be well-suited to GIC's. Perhaps if high-stage superconducting GIC's are ever discovered, it might be worthwhile to take another look at proximity-coupling. For the known GIC's, though, it is imperative to consider alternative theories.

Even though intuition says that the enhancement of the critical fields in superconducting GIC's is linked to their anisotropy, it is important to consider whether mechanisms not related to anisotropy could also play a role. The major causes of critical field enhancement in isotropic superconductors are spin-orbit scattering and strong-coupling behavior.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995