One often hears the statement that large spin-orbit scattering
increases Hc2(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 Hc2 in
C4KHg. 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 Hc2(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 Hc2(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 Hc2. 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 rhon
is the normal-state dc resistivity. In Eqn. , m is the bare
electron mass, not the effective mass, since it comes from ehbar/2 muBc.[53] For C4KHg, the best method is to use rhoa to calculate alpha for vecH || ^c, and sqrtrhoa rhoc to calculate alpha for vecH _|_ ^c. The
parameters used were rhoc = 0.2 milliohm-cm at 4.2
K,[85] and Gamma = 0.95 millijoules / (mol
K2)[8]. rhoa at 4.2 K for C4KHg (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 rhoa data
exist.[70] Values of alpha obtained from both halves of
Eqn.
are given in Table
,
which demonstrates that C4KHg 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 C4KHg.
Table: Comparison for C4KHg 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 Lambdaep, the same
electron-phonon parameter that appears in the dressed density-of-states
found from specific-heat measurements.[10] Note that Lambdaep is not related to the magnetic-field penetration depth
Lambda (see Section for a discussion of the penetration
depth). McMillan[165] calculated Tc in terms of
Lambdaep and mu *, the Coulomb pseudopotential of Morel and
Anderson[174]. He found:
where thetaD is the Debye temperature. Using a typical number of mu* = 0.1,[165], one can solve for Lambdaep:
The results of this calculation for the superconducting GIC's
whose Debye temperature has been measured are collected in
Table . Lambdaep 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 Tc of about
1-2 K.
Table: Values of Lambdaep, the electron-phonon coupling
parameter, for GIC superconductors. Tc = 0.73 K[120] is used
for C4KHg since no transition was observed down to 0.8 K during the
specific-heat measurement.[8] Values of Lambdaep
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 Lambdaep 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 Lambdaep >= 1. Furthermore, since the strong-coupling enhancement is larger near Tc 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 Hc2(t). Bulaevskii and Dolgov[31] find that h*(0) = 0.45sqrtLambdaep for Lambdaep >> 1. Marsiglio and Carbotte find h*(0) about 1.6, but only when Tc thetaD.[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 C4KHg. 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 C4KHg. 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 Dave is the mean diffusivity, and y ==
D/ Dave.
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) == DaveP(D), where P(D) is the
distribution function for diffusivity, Dave 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 Tc, 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, Hc2 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 C4KHg,
but it also has some problems. One is that among the GIC superconductors,
multiphase behavior has been observed only in C4KHg. As is
discussed in Chapter , C4KHg is remarkable among the
superconducting GIC's for the wide range of Tc'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 C4KHg 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, C4KHg 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 Lambdatr == 0.882 xi0 / l.[108] Here xi0 == 0.18 hbar vF/ kB Tc is the Pippard coherence length, and
l is the mean-free path. For C4KHg, xi0 is roughly 9000
Å, and la, 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
la xi0, the dimensionless parameter Lambdatr == 0.88 xi0/l 0.86 for in-plane transport. Lambdatr < 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 vecH || ^c. However, this calculation does not rule out a
diffusivity-variation influence on vecH _|_ ^c. Transport
is expected to be much dirtier along ^c, where the resistivity
is about 300 times higher than in-plane.[85] Lambdatr 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 Hc2 can be justified for C4KHg for vecH _|_ ^c, but it is hard to justify for vecH || ^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 C4KHg, they probably do not have a dominant effect on Hc2(T).
Finally, to round out the discussion of critical field enhancement
in isotropic superconductors, mention should be made of the anomalous Hc2(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 dHc2/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 Hc2, 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(Hc2) / rho.[57] Here Delta rho(Hc2) / rho is
the magnetoresistance at the field Hc2; the normal state
resistivity at zero-field is extrapolated from above Tc. 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(Hc2)/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 vecH _|_ ^c, but except for C4KHg, there is no hard evidence for multiphase behavior in other superconducting GIC's except for C4KHx.[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.