One often hears the statement that large spin-orbit
scattering increases H_{c2}(T) in a high-field
superconductor. Since the spin-orbit interaction increases
rapidly as a function of the atomic number[170] Z, and since Hg is a
high-Z element, it is logical to ask whether spin-orbit
scattering could enhance H_{c2} in C_{4}KHg.
The answer is no, since it turns out that the spin-orbit
``enhancement'' only affects materials whose critical fields
have already been depressed by Pauli limiting effects.[262,186] ``Pauli limiting'' refers to
the upper bound on H_{c2}(T) from spin-susceptibility
(Pauli paramagnetism) effects.[45,38] Superconductors close
to their Pauli limit have critical fields which are depressed
from the Maki-deGennes temperature dependence. Paramagnetic
limiting also affects the angular dependence of the critical
field.[9,53] High rates of spin-orbit
scattering can increase the susceptibility of the
superconducting state to that of the normal state, and thus
eliminate Pauli-limiting effects. Thus spin-orbit scattering
and Pauli limiting effectively cancel one another out, and
the final result of both effects is the same Maki-deGennes
curve already discussed in Section
.[186] Since the two
mechanisms cancel, and since spin-orbit scattering has no
major impact for a superconductivity far from the
paramagnetic limit, spin-orbit scattering cannot give a
reduced field h^{*}(0) **>** 0.7, and cannot
explain the enhanced linearity of H_{c2}(T) observed
in GIC's.

While on the subject of electron spin, it is worth mentioning
that GIC's appear to be far from the paramagnetic limit, as
would be expected for low-critical field materials. The
reason is that orbital pair-breaking effects are strong
enough in GIC superconductors that they are far from the
spin-susceptibility ceiling on H_{c2}. The small
values of the Maki *alpha* parameter[159] for GIC's clearly demonstrate
the validity of this statement, since spin contributions to
the energy of a superconductor become important when
*alpha* >= 1. WHH[262] provided two ways of
calculating *alpha*:

Here *Gamma* is the linear specific heat coefficient,
and *rho*_{n} is the normal-state dc
resistivity. In Eqn. , m is the bare electron
mass, not the effective mass, since it comes from
e*hbar*/2 *mu*_{B}c.[53] For C_{4}KHg, the
best method is to use *rho*_{a} to calculate
*alpha* for *vec*H || ^c, and
*sqrt**rho*_{a} *rho*_{c} to
calculate *alpha* for *vec*H _|_ ^c. The parameters
used were *rho*_{c} = 0.2 milliohm-cm at 4.2
K,[85] and *Gamma* =
0.95 millijoules / (mol K^{2})[8]. *rho*_{a} at
4.2 K for C_{4}KHg (which has not been reported) was
estimated by assuming a resistivity anisotropy of 280, the
same as the measured resistivity anisotropy at 100 K, the
lowest temperature at which published *rho*_{a}
data exist.[70] Values
of *alpha* obtained from both halves of Eqn.
are given in Table , which demonstrates that
C_{4}KHg is two to three orders of magnitude away
from Pauli-limiting regime. With like assumptions, similar
calculations for other GIC's give *alpha*'s of the same
order of magnitude. Therefore spin-orbit scattering has no
impact on the critical fields of C_{4}KHg.

**Table:** Comparison for C_{4}KHg of
two different methods for determination of Maki's
*alpha* parameter.[262] The orientation indicated
is that of the applied magnetic field. In parentheses it is
noted which of the two halves of Eqn.
was used.

WHH say about Eqn. that ``It is a test of
the applicability of our model for the superconductor that
these two determinations of *alpha* should agree.''[262] Considering the
crudeness of the assumptions made in the estimation of
*rho*, the agreement in Table
between the two determinations of *alpha* is quite good.
This is an indication that even though superconductivity in
GIC's has some anomalous aspects, it still is explained by
the same basic electron-phonon coupling mechanism that
applies to isotropic superconductors. Therefore, in our
search for models to explain the enhanced critical fields of
GIC's, models with truly exotic coupling schemes (
*e.g.,* plasmons and excitons) can safely be ignored.

Another factor that contributes to critical field enhancement
in isotropic superconductors besides spin-orbit scattering is
strong-coupling effects. ``Strong-coupling'' refers to the
case of a large electron-phonon interaction. The strength of
electron-phonon coupling is measured by the size of the
dimensionless parameter *Lambda*_{ep}, the same
electron-phonon parameter that appears in the dressed
density-of-states found from specific-heat measurements.[10] Note that
*Lambda*_{ep} is not related to the
magnetic-field penetration depth *Lambda* (see Section
for a discussion of the penetration depth). McMillan[165] calculated
T_{c} in terms of *Lambda*_{ep} and
*mu* ^{*}, the Coulomb pseudopotential of Morel
and Anderson[174]. He
found:

where *theta*_{D} is the Debye temperature.
Using a typical number of *mu*^{*} = 0.1,[165], one can solve for
*Lambda*_{ep}:

The results of this calculation for the superconducting GIC's
whose Debye temperature has been measured are collected in
Table .
*Lambda*_{ep} in GIC superconductors appears to
be 0.4, about the same as in prototypical weak-coupling
superconductors like aluminum and zinc.[165] This is in keeping with
one's expectations for a material with a rather low
T_{c} of about 1-2 K.

**Table:** Values of
*Lambda*_{ep}, the electron-phonon coupling
parameter, for GIC superconductors. T_{c} = 0.73 K[120] is used for
C_{4}KHg since no transition was observed down to 0.8
K during the specific-heat measurement.[8] Values of
*Lambda*_{ep} for the KH-GIC's are gathered in
Table .

In amorphous superconductors, strong-coupling effects can
cause extended linearity at low temperatures.[19] However, the values of
*Lambda*_{ep} quoted above eliminate the
possibility that GIC superconductors are subject to any
critical field field enhancement from strong-coupling
effects, since these effects are important only for
*Lambda*_{ep} >= 1. Furthermore, since the
strong-coupling enhancement is larger near T_{c} than
at low t,[53,50] strong-coupling effects
tend to actually *decrease* h^{*}(0), the
reduced field at zero temperature.

Recently some theories of ultra-strong coupling
superconductivity have been published, inspired by the advent
of high-temperature superconductivity.[31,162] Some of these models find
positive curvature of H_{c2}(t). Bulaevskii and
Dolgov[31] find that
h^{*}(0) = 0.45*sqrt**Lambda*_{ep}
for *Lambda*_{ep} >> 1. Marsiglio and
Carbotte find h^{*}(0) about 1.6, but only when
T_{c} *theta*_{D}.[162] These models are clearly
not applicable to superconductivity in known GIC's.

For GIC's, a more relevant consideration than strong-coupling
is inhomogeneity. From both structural[123,246] and superconducting[206,55] studies, there is abundant
evidence for the coexistence of multiple phases in
C_{4}KHg. This evidence is discussed in detail in
Chapter . As far as critical
field experiments go, this multiphase behavior is important
because of the possibility that inhomogeneity is the cause of
the enhanced linearity of the critical fields of
C_{4}KHg. Carter and coauthors[34] developed a model for the case
of multiphase materials which contain both an equilibrium and
higher free-energy phase. The model treats the inhomogeneity
by allowing the superconductor to have a distribution of
diffusivities described by a function P(D). Then, instead of
the Maki-deGennes equation (Eqn. )
for the critical fields of dirty superconductors, one uses:

where all the symbols are the same as before, and *psi*
is the digamma function. Carter *et al.* found that by
widening the distribution P(D) from a *Delta*-function
(implied by the choice of a single D value) to a broad hump
that they could produce both positive curvature and
h^{*}(0) 0.85. By skewing the distribution P(D) to
low D, they could even get h^{*}(0) **>**
0.9.[34] The results of
their calculations are shown in Fig. ,
where h^{*} (called h in the axis label) is plotted
versus t as a function of the normalized diffusivity
distribution function, Q(y). Q(y) == D _{ave} P(D),
where D_{ave} is the mean diffusivity, and y == D/
D_{ave}.

**Figure:** Extended critical field linearity
due to small-scale sample inhomogeneity. From a calculation
by Carter and colleagues.[34] The plots are of reduced field
versus reduced temperature for several different normalized
diffusivity distribution functions Q(y). Q(y) ==
D_{ave}P(D), where P(D) is the distribution function
for diffusivity, D_{ave} is the average diffusivity,
and y == D/D _{ave}. In the lower plot, a P(D) skewed
to lower diffusivities produces an even greater critical
field enhancement at low temperatures. The index n refers to
the power of the linear factor used to skew the symmetric
distribution.

The physical cause of the inhomogeneity-related enhancement
has to do with the temperature dependence of *xi*, which
is the approximate radius of a normal vortex in a
superconductor.[252] At T
T_{c}, *xi* is large, so that vortices must
extend over both high-D and low-D regions in the material. At
low t, where *xi* has grown considerably smaller, the
material can save some condensation energy by preferentially
packing the vortices into the low-D regions with higher
critical fields. As a result, when *xi*(T) becomes on
the order of the domain size, H_{c2} will turn
upward.[34]

Does this model offer an explanation of positive curvature
and enhanced linearity in the critical fields of GIC's?
Clearly the inhomogeneity interpretation has some appealing
features for C_{4}KHg, but it also has some problems.
One is that among the GIC superconductors, multiphase
behavior has been observed only in C_{4}KHg. As is
discussed in Chapter , C_{4}KHg is
remarkable among the superconducting GIC's for the wide range
of T_{c}'s it exhibits (from 0.7 to 1.6 K), and
because it undergoes what is apparently an ordering
transition under the influence of small perturbations
(hydrogenation[206] and small
hydrostatic pressure[55]).
These features are not observed for other GIC's, which have
well-defined transition temperatures and show no unusual
behavior under pressure.[116,55] It does not seem to make sense
to attribute the anomalies in C_{4}KHg to a different
cause than the deviations seen in other GIC's, especially
considering that the other GIC's show larger anomalies (see
Fig.
).

Even if one were willing to assume separate causes for the
enhanced critical fields of the various GIC's, it is not
clear that the model of Carter and colleagues would be
applicable. The problem is that their model makes the
(reasonable) assumption that microscopically inhomogeneous
superconductors will be in the dirty limit, where the
Maki-deGennes equation is applicable. However,
C_{4}KHg appears to be fairly ``clean,'' at least for
in-plane transport. The standard way to quantify cleanliness
in a superconductor is to calculate the parameter
*Lambda*_{tr} == 0.882 *xi*_{0} /
*l*.[108] Here
*xi*_{0} == 0.18 *hbar* v_{F}/
k_{B} T_{c} is the Pippard coherence length,
and *l* is the mean-free path. For C_{4}KHg,
*xi*_{0} is roughly 9000 Å, and
*l*_{a}, the in-plane mean-free path, is about
9100 Å. (These numbers were obtained from
Shubnikov-deHaas data[245]
using standard rigid-band analysis, as demonstrated in
Appendix .) Since
*l*_{a} *xi*_{0}, the dimensionless
parameter *Lambda*_{tr} == 0.88
*xi*_{0}/*l* 0.86 for in-plane transport.
*Lambda*_{tr} **<** 1 is indicative of
fairly clean behavior,[108] so this is an indication
that inhomogeneities are not likely to be the cause of
enhanced linearity, at least for *vec*H || ^c. However,
this calculation does not rule out a diffusivity-variation
influence on *vec*H _|_ ^c. Transport is expected to be
much dirtier along ^c, where the resistivity is about 300
times higher than in-plane.[85] *Lambda*_{tr} is
estimated very roughly to be about 50 for c-axis transport by
assuming a spherical band. (No well-bounded number is
available since the Shubnikov-deHaas data give no information
about the intercalant bands.[245] c-axis transport in GIC's is
discussed in Section .)

In summary, use of the inhomogeneity model for enhanced
H_{c2} can be justified for C_{4}KHg for
*vec*H _|_ ^c, but it is hard to justify for *vec*H
|| ^c, or for other superconducting GIC's. Therefore the most
sensible conclusion is that, while the factors discussed by
Carter *et al.* may play a role in C_{4}KHg,
they probably do not have a dominant effect on
H_{c2}(T).

Finally, to round out the discussion of critical field
enhancement in isotropic superconductors, mention should be
made of the anomalous H_{c2}(T) behavior found in
heavy-fermion superconductors.[57] These unusual materials
exhibit both positive curvature and large values of
h^{*}(0). Any explanation of these phenomena must
remain tentative since the basic physics of these compounds
is still controversial,[156]
but recently an interesting model has been proposed by DeLong
*et al.*[57] The
explanation is based on the observation that
dH_{c2}/dT *propto* *rho* in the usual WHH
model, as shown in Eqn. . If the normal-state
resistance *rho* is assumed independent of field up to
H_{c2}, then one gets the usual WHH result
h^{*}(0) 0.7. However, if there is a very strong
magnetoresistance (the manifestation of a field-dependent
pairing interaction), then the slope formula shows that
h^{*}(0) can exceed 0.7 by a factor of *Delta*
*rho*(H_{c2}) / *rho*.[57] Here *Delta*
*rho*(H_{c2}) / *rho* is the
magnetoresistance at the field H_{c2}; the normal
state resistivity at zero-field is extrapolated from above
T_{c}. This formalism may have wide applicability to
superconductors with large or anomalous magnetoresistance;
however, it appears unlikely to help much in the case of
GIC's, since the critical fields are small enough that the
magnitude of *Delta*
*rho*(H_{c2})/*rho* is anticipated to be 0.
The magnetoresistance of the KHg-GIC's was observed by Timp
and coworkers, who reported nothing extraordinary.[245]

Four causes of critical field enhancement in isotropic
superconductors have been discussed: spin-orbit scattering,
strong-coupling effects, inhomogeneities, and
magnetoresistance. Of these, only the
multiphase-superconductor model of Carter *et al.* is
thought to be relevant to GIC superconductors. Inhomogeneity
effects could play a role for *vec*H _|_ ^c, but except
for C_{4}KHg, there is no hard evidence for
multiphase behavior in other superconducting GIC's except for
C_{4}KH_{x}.[79,232]

The extremely anisotropic nature of superconductivity in
GIC's (1/*epsilon* as high as 47[116]) provides motivation to consider
models of positive curvature and enhanced linearity which
treat orientation-dependent effects as central. Some of these
models are considered in the next section.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995