Many of the experimental factors relevant to the angular
behavior of the critical field will also complicate the
temperature dependence. Among these are sample misalignment,
mosaic spread, and critical field definition. In addition
there are the considerations of temperature measurement,
stability, and equilibration. (See Section
.) Nonetheless, the H_{c2}(T) results are a bit
easier to interpret than the H_{c2}(*theta*)
ones, if only because the functional form of the data is
simpler. The approximately linear behavior of the temperature
dependence curves is illustrated in Figure ,
which shows H_{c2}(T) curves for four different
C_{4}KHg GIC's. These data are from the same samples
whose H_{c2}(*theta*) curves are shown in Figure
.

**Figure:** Critical field H_{c2} as a
function of reduced temperature for C_{4}KHg. Dotted
curves are least-squares line fits to the data. Fit
parameters are given in Table .
a) Data for a C_{4}KHg with T_{c} = 0.95 K:
(*circ*), *vec*H _|_ ^c. (*bullet*),
*vec*H || ^c Data for a T_{c} = 0.73 K sample
from Ref. [240]:
(*diamond*), *vec*H _|_ ^c. (×), *vec*H
|| ^c. b) Data for two C_{4}KHg-GIC's with
T_{c} 1.5 K. T_{c} = 1.53 K sample:
(*circ*), *vec*H _|_ ^c. (*bullet*),
*vec*H || ^c. T_{c} = 1.54 K sample:
*diamond*, *vec*H _|_ ^c. ×, *vec*H ||
^c.

The fits in Figure are all to the simple
linear form

where t is the reduced temperature, T/ T_{c}. The
least-squares-determined values of H_{c2}(0) and
T_{c} for these curves are given in Table .
The quality-of-fit parameter *cal* R is calculated as in
Eqn. , only here the error at
the **i**th point *sigma*_{i} is taken to be
the same as the experimental field value, that is:

Of course the errors should really be smaller in magnitude
than the data, but their fractional size is thought to be
roughly independent of temperature, the independent variable.
Letting the estimated error be the same as the data is a
convenient way of implementing this standard assumption,
which is called statistical weighting.[21] The field error bars shown
on the plot are twice the mean error per data point in total
length, where the mean error *sigma* is given by

where *nu* is the number of degrees of freedom. This is
a standard way of defining error bars[202] which includes the effects only
of random errors, not systematic ones. The temperature error
bars are based on an uncertainty of 40 mK at the low end, and
10 mK at the high end, for reasons discussed in Section
.

**Table:** Parameters obtained from
least-squares fits of Eqn.
to C_{4}KHg H_{c2}(T) data. The individual
specimens are labeled by the value of T_{c} obtained
from the zero-field sweeps. The other parameters, including
the ``fit'' T_{c}, are obtained from the
least-squares linear fits to the critical field data that are
shown in Figure . The residual
*cal* R is calculated using Eqn. .
NR denotes that the parameter was not reported in the cited
reference.

In general, the data in Figure
are quite well described by Eqn. .
Ginzburg-Landau models predict a linear
temperature-dependence of H_{c2} in their region of
validity, so this result is not surprising, especially since
Iye and Tanuma[120] found
linear temperature dependence for their C_{4}KHg
samples. One feature of Table
that merits some discussion is that the T_{c}
determined from the best linear fit is as much as 10% higher
than that determined from the zero-field temperature sweeps.
This disparity between the different ways of measuring
T_{c} is a fairly common occurrence in
superconductivity which is generally caused by curvature of
the H_{c2}(T) data in the region near
T_{c}.[186]
Hohenberg and Werthamer[111] found that Fermi-surface
anisotropy can cause positive curvature of the critical field
near T_{c}, but here negative curvature is needed in
order to explain the high extrapolated values of
T_{c}.

For the higher- T_{c} specimens, the T_{c}
numbers found from the *vec*H || ^c fits are higher than
those found from the *vec*H _|_ ^c fits. Because of the
discussion of type I superconductivity in the Section ,
one might wonder if the H_{c2, || ^c}(T) data in
these samples are actually quadratic, since a quadratic
temperature dependence is expected for measurements of the
thermodynamic critical field. A moment's thought indicates
that a linear fit to a quadratic function will have a higher
intercept on the x-axis than a quadratic fit will. (Se Figure
.) Therefore, if the data are truly quadratic, it is
easy to see why a linear fit will give a falsely high
T_{c}. The temperature dependence data for the
T_{c} 1.5 K GIC's for *vec*H ||^c are shown on
an expanded scale in Figure ,
along with the best quadratic and linear fits. This plot is
quite similar to Figure , where the presumed
thermodynamic critical field values were obtained from fits
to H_{c2}(*theta*). Here the data were obtained
directly, using field sweeps at constant *theta* as a
function of temperature.

**Figure:** Critical fields with *vec*H ||
^c for T_{c} 1.5 K C_{4}KHg samples.
*Uparrow* marks the value of T_{c} found using a
zero-field temperature sweep. a) (*bullet*), data for a
T_{c} = 1.53 K sample; (*diamond*), a linear fit
to the data with H_{c2}(0) = 89.7 Oe, T_{c} =
1.65 K and *cal* R = 6.25e-3; (.), a quadratic fit to
the data with H_{c2}(0) = 64.0 Oe, T_{c} =
1.55 K and *cal* R = 1.2e-2. b) (*bullet*), data
for a T_{c} = 1.54 K sample; (*diamond*), a
linear fit to the data with H_{c2}(0) = 85.8 Oe,
T_{c} = 1.62 K and *cal* R = 1.62e-3; (.), a
quadratic fit to the data with H_{c2}(0) = 62.8 Oe,
T_{c} = 1.51 K and *cal* R = 4.7e-2.

Figure shows that using a
quadratic function to fit the *vec*H || ^c data does
improve the agreement between the fit T_{c} and that
obtained from a zero-field sweep. Unfortunately, the
quadratic function does not describe the data as well as a
linear function: for one sample the residual for the
quadratic fit is twice that of the linear fit, and for the
other it is three times that of the linear fit. Consultation
of standard tables[21]
shows that the difference between these fits has only about
an even chance of being significant (for about 15 data
points). Therefore the only safe statement is that the
question of type I superconductivity in C_{4}KHg is
still unresolved, since the fits to H_{c2, || ^c} are
inconclusive. As was mentioned in relation to Figure ,
in C_{8}K both H_{c}(T) and H_{c2,_|_
^c}(T) show positive curvature,[141] so perhaps in C_{4}KHg
it is not implausible for both critical fields to show a
linear temperature dependence.

Some lower- T_{c} samples, such as the T_{c}
= 0.95 K specimen whose data is shown in Figure
(and others whose data are not displayed) showed
T_{c}'s which were higher than the zero-field values
for both field orientations. This fact suggests that some
experimental error is causing the discrepancies. Because the
*vec*H || ^c measurements were usually taken going down
in temperature, and those for *vec*H _|_ ^c were
generally taken going up in temperature (see Section
), it appears unlikely that systematic temperature
measurement problems are responsible. At this time the high
values of T_{c} obtained from the H_{c2, ||
^c} fits remain unexplained, but it is felt that some
type of measurement problem is the most likely cause.

Another unexpected feature of the data for both orientations
is that the linearity of H_{c2}(t) persists to
unusually low reduced temperatures. For typical type II
superconductors, Eqn. holds only for about 0.6
**<** t **<** 1.0, below which a saturation in
H_{c2}(t) becomes noticeable. This saturation is
described quantitatively by the widely accepted theory of
Werthamer, Helfand, and Hohenberg (WHH)[262] which has been successful
in describing a wide variety of superconducting materials[186,84]. Anisotropic materials which
are described by the AGL model for their angular dependence
are expected to have a temperature dependence at constant
angle describable by the WHH theory. The basic equation of
this theory is one developed by Maki and deGennes[210,108]:

where t is the reduced temperature, D is the diffusivity, and
the digamma function *psi*(x) is related to the gamma
function by:

This equation is strictly applicable only to dirty
superconductors or clean superconductors near T_{c},
but the theory has been extended to lower temperatures for
clean superconductors by Helfand and Werthamer[108]. The contribution of WHH[262] was to further extend
the theory to incorporate effects due to Pauli paramagnetism
and spin-orbit coupling. The meaning and possible importance
of these embellishments is discussed below in Section
. Simple two-parameter ( T_{c}, <=ft.
dH_{c2}/dT *|*_{T}c) WHH fits to the
C_{4}KHg data are shown in Figure ,
where they are compared to the best linear fits, the same
linear fits shown in Figure .

**Figure:** Comparison of WHH and linear fits
to H_{c2}(T) data taken on a T_{c} = 1.54 K
sample. a) (*bullet*), data for *vec*H _|_ ^c. (.),
linear fit with H_{c2}(0) = 748 Oe, T_{c} =
1.52 K, and *cal* R = 6.9e-3. (*circ*), WHH fit
with H_{c2}(0) = 518 Oe, T_{c} = 1.53 K, and
*cal* R = 1.6e-2. b) (*bullet*), data with
*vec*H || ^c. (.), linear fit with H_{c2}(0) =
85.8 Oe, T_{c} = 1.62 K, and *cal* R = 1.6e-3.
(*circ*), WHH fit with H_{c2}(0) = 59.76 Oe,
T_{c} = 1.63 K, and *cal* R = 1.2e-2.

As might be expected, the Maki-deGennes equation gives
approximately linear behavior near T_{c}. Therefore
the linear and WHH fits are in good agreement just below
T_{c}. However, the low-temperature extension[108] of the Maki-deGennes
formalism produces the result

This formula is the quantitative expression of the saturation shown in the WHH curves of Figure , saturation which the data does not appear to exhibit. The better agreement of the linear fits than the WHH fits is confirmed by the residual indices, which are significantly lower for the linear fits.

A more impressive demonstration of linearity is shown in
Figure , which gathers together
the data of 5 samples for both orientations. Here the data
are plotted in dimensionless units on both axes; the reduced
field h^{*} is defined by

For these 143 data points, the residual index for the linear
fit is 3/4 that of the WHH fit, indicative of about a 90 %
probability that the line describes the data better. Figure
also shows that at the lowest reduced temperature the
data have already reached h^{*} 0.7, the
zero-temperature value of h^{*} calculated using the
Helfand-Werthamer[108]
formalism. Therefore, while it would be gratifying to measure
H_{c2}(t) to lower reduced temperatures and see even
larger deviations from the WHH theory, the available data
(down to t = 0.3) demonstrate convincingly that the observed
deviations are real.

**Figure:** Summary of all H_{c2} data,
both _|_ and || to the c-axis. The dimensionless quantities
plotted are reduced field ( h^{*}) versus reduced
temperature (t). (*bullet*), 143 data points taken on 5
different GIC's. (*circ*), best 2-parameter WHH fit to
the data with *cal* R = 1.7e-2. (.), best linear fit to
the data with *cal* R = 1.3e-2. Both fits have
*frac*dh^{*}dt = -1 at t = 1.

The conclusions drawn here are perfectly consistent with
those stated by Iye and Tanuma in their papers on
C_{4}KHg.[120] The
reason is that their data extended over a smaller temperature
range than the MIT data. At the lowest reduced temperature
for which they reported measurements on C_{4}KHg, t =
0.55, linear character is consistent with both the WHH and
linear fits, as Figure shows. Therefore
extended linearity was unobservable in Iye and Tanuma's
samples in the temperature interval in which they performed
measurements. It should be noted, though, that unusual
H_{c2}(T) behavior may not occur in the lower-
T_{c} samples. Considering that the
H_{c2}(*theta*) for the lower- T_{c}
GIC's were well-described by the simple AGL model, it seems
reasonable that the lower- T_{c} H_{c2}(T)
curves may agree well with the Maki-deGennes equation.

Examination of Equation
shows that there is a lot more information that can be
extracted from the H_{c2}(T) data. To begin with, the
value of *epsilon*, the anisotropy parameter,[175] can be calculated as

and compared with the *epsilon* obtained from fits to
the H_{c2}(*theta*) data. The resultant numbers
are displayed in Table .

**Table:** Comparison of the anisotropy
parameter *epsilon* as obtained from H_{c2}(T)
and H_{c2}(*theta*) fits. The H_{c2}(T)
*epsilon* numbers were obtained from the ratio of the
slopes. The H_{c2}(*theta*) numbers were
obtained from fits using Eqns.
(AGL) and (Tinkham's formula),
with type I superconductivity allowed for small *theta*.
In each case, the TF fits had lower residuals than the AGL
fits (see Table ). NA indicates that a
TF fit was not performed on this data; NR denotes information
that was not reported in the cited reference.

As the table shows, agreement between the two methods of
determining *epsilon* is rather poor. While
*epsilon* as determined from H_{c2}(T) is
consistently about 9, that determined from
H_{c2}(*theta*) ranges from 8 to 16. The reason
that the angular dependence's *epsilon*'s are higher is
that in fitting H_{c2}(*theta*) it has been
assumed that the number measured for H_{c2}(0°)
is actually H_{c}, and that the real value of
H_{c2}(0°) is much lower. As discussed in Section
, the H_{c2}(*theta*) data cannot be fit
without making use of this assumption. In addition, the
presence of type I superconductivity is supported by the
C_{v} measurements done by Alexander *et
al.*[8]

It is tempting to conclude that the values for *epsilon*
determined from the angular dependence are unreliable because
of problems with the fits. The matter is not that simple
though, since the variability of *epsilon* can be
demonstrated in a model-independent way using the
H_{c2}(*theta*) data, without benefit of fits.
One way of doing this is to plot
H_{c2}(*theta*)/H_{c2}(0°) versus
*theta*. This scaling forces the curves through the
common point (0°, 1), with the value at the peak being
1/*epsilon* at the temperature of measurement. If
*epsilon* is in fact constant with respect to
temperature, then the
H_{c2}(*theta*)/H_{c2}(0°) versus
*theta* curves should all overlay one another except for
random errors.

**Figure:** Demonstration of the temperature
dependence of the anisotropy parameter *epsilon* in
C_{4}KHg, where 1/*epsilon* ==
H_{c2}(90°)/H_{c2}(0°). Data are for
a T_{c} = 1.54 K C_{4}KHg sample.
(*circ*), t = 0.29. (*bullet*), t = 0.55.
(×), t = 0.76. All H_{c2}(0°) values were
determined from the data, not the fits, so that this plot is
model-independent. Fits to this data are shown in Figure
.

That this is not the case is demonstrated in Figure .
The t = 0.55 and t = 0.76 curves appear to overlay for the
most part, but the t = 0.29 curve clearly rises above the
other two at the peak. Figure
therefore suggests that the variation in *epsilon*
between t = 0.29 and t = 0.55 is real, but that any variation
between t = 0.55 and t = 0.76 is at best small. From Table ,
one can see that the magnitudes of *epsilon* from the
best fits (to Tinkham's formula) support this conclusion,
although the verdict of the higher-residual AGL fits is less
clear.

A temperature-dependent *epsilon* implies a
temperature-dependent slope in at least one of the
high-symmetry directions. To be more precise, an increase in
the anisotropy at low reduced temperatures requires either a
downward deviation from linearity in H_{c2, || ^c} or
an upward deviation from linearity in H_{c2, _|_ ^c}.
While it is certainly true that neither of these trends is
obvious in the H_{c2}(T) data, it also turns out that
neither of these is ruled out. The reason is that if the idea
of type I superconductivity in C_{4}KHg is taken
seriously, then the data plotted in Figure
b) are H_{c} rather than H_{c2}, and so make
no statement about any hypothetical curvature of H_{c2, ||
^c}(T). Any positive curvature of H_{c2, _|_ ^c}
would be lost in the noise of the data at low reduced
temperatures, so that any kink of the size predicted by the
H_{c2}(*theta*) would be unobservable.

So what are the best numbers for *epsilon*(T) from the
sum total of these data sets? One important point to notice
is that the magnitude of the anisotropy derived from the
linear fits to the H_{c2}(T) curves represents a sort
of thermal average. Thus *epsilon* = 9 is probably the
mean value over the range 0.3 **<** t **<** 0.95.
The evidence from the angular dependence, which measures the
anisotropy at a specific t, indicates that the true
*epsilon* is higher at low reduced temperatures and
lower at higher reduced temperatures. Because the
H_{c2}(*theta*) data is much less affected by
experimental errors, the magnitudes of *epsilon*
determined from the H_{c2}(*theta*) fits are
thought to be more reliable.

Temperature-dependent anisotropy has been observed before in
graphite intercalation compounds, specifically in
C_{8}KHg by Iye and Tanuma. The increase in
anisotropy from 17.6 at t = 0.81 to 21.6 at t = 0.23 is
illustrated in Figure . The variation of
*epsilon* shown there is similar to what is reported
here for C_{4}KHg. C_{8}KHg is the only GIC
for which Iye and Tanuma showed H_{c2}(*theta*)
curves at more than one temperature. This is unfortunate,
since it would be interesting to know whether a variable
*epsilon* is a general property of GIC superconductors.

**Figure:** Temperature-dependent anisotropy in
C_{8}KHg is demonstrated by a plot of
H_{c2}(*theta*)/H_{c2}(0°) versus
*theta*, just as in Figure .
All data from Iye and Tanuma, Ref. [240] on a T_{c} = 1.94 K
sample. (×), data at t = 0.23. Fit, (*diamond*),
with 1/*epsilon* = 17.6 and *cal* R = 6.8e-3.
(*bullet*), data at t = 0.81. (*circ*), fit with
1/*epsilon* = 21.6 and *cal* R = 5.3e-3.

**Figure:** Positive curvature of
H_{c2}(T) in C_{8}RbHg. Data are taken from
Iye and Tanuma, Ref. [120].
(*circ*), H_{c2, _|_ ^c}. (*bullet*),
H_{c2, || ^c}. Parameters for the line fits: for
*vec*H _|_ ^c, H_{c2}(0) = 3078 Oe,
T_{c} = 1.36 K, and *cal* R = 0.56; for
*vec*H || ^c, H_{c2}(0) = 89.0 Oe, T_{c}
= 1.37 K, and *cal* R = 3.02e-2. Zero-field
T_{c} for this sample was 1.4 K.[120]

There is some justification for speculating that a
temperature-dependent anisotropy is a common property of the
class. According to data shown in Ref. [120], all of the superconducting
GIC's (with the possible exception of C_{4}KHg) show
positive curvature of their upper critical fields with
respect to temperature (d
H_{c2}^{2}/dT^{2} **>** 0). The
largest curvature seems to occur in C_{8}RbHg, as the
data in Figure show. Positive curvature
does not guarantee a variable *epsilon*, of course, but
the chances of exactly the same curvature in both critical
fields seems small. In terms of the theories described in
Section , identical curvature in
both of the orientations would be a coincidence rather than a
natural consequence of the models (although identical
curvature for both orientations has been reported for at
least one compound, Fe_{0.05}TaS_{2}[46]). These models are discussed
in more detail in the section that follows.

In the section that follows, reference will frequently be
made to theories which predict positive curvature of
H_{c2}(T). These references should not be understood
as suggestions that positive curvature is observed in
C_{4}KHg, but rather as expressions of the philosophy
that whatever phenomena are the cause of the positive
curvature in C_{8}RbHg are most likely also to cause
the extended linearity seen in C_{4}KHg. Theories
which can produce upward-curving critical field curves should
easily be able to produce straight ones through adjustment of
parameters. In these models, after all, a straight
H_{c2}(T) means merely the compensation of the forces
which drive upward and downward curvature.

Before moving on to the interpretation of these results,
let's pause to summarize. In C_{4}KHg, the critical
field both perpendicular and parallel to the graphite c-axis
shows enhanced linearity with respect to the usual theory for
type II superconductors. The data parallel to ^c are only
marginally consistent with the quadratic behavior expected
for the thermodynamic critical field of type I
superconductors. However, H_{c}(T) in C_{8}K
is not well-fit by a quadratic, either. Clem has shown that
superconducting energy gap anisotropy can cause a slight
deviation from the BCS temperature dependence of
H_{c}.[44] However,
the deviation expected in Clem's theory is so small that his
model should not be considered a serious candidate to explain
the deviations seen in GIC's. The introduction of the
possibility of type I superconductivity was suggested by the
specific heat data of Alexander *et al.*, and also by
the poor quality of the H_{c2}(*theta*) fits
without it. The H_{c2}(*theta*) fits also
suggest that the anisotropy parameter *epsilon* is
temperature-dependent. Both the temperature dependence of
*epsilon* and the enhanced linearity of
H_{c2}(T) are inexplicable within the simple
anisotropic Ginzburg-Landau theory,[244,155] although basically the data
are well-described by this model. The AGL and other, more
detailed, models are discussed at length in the next section.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995