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 Hc2(T) results are a bit
easier to interpret than the Hc2(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 Hc2(T) curves for four different
C4KHg GIC's. These data are from the same samples
whose Hc2(theta) curves are shown in Figure
.
Figure: Critical field Hc2 as a
function of reduced temperature for C4KHg. Dotted
curves are least-squares line fits to the data. Fit
parameters are given in Table .
a) Data for a C4KHg with Tc = 0.95 K:
(circ), vecH _|_ ^c. (bullet),
vecH || ^c Data for a Tc = 0.73 K sample
from Ref. [240]:
(diamond), vecH _|_ ^c. (×), vecH
|| ^c. b) Data for two C4KHg-GIC's with
Tc 1.5 K. Tc = 1.53 K sample:
(circ), vecH _|_ ^c. (bullet),
vecH || ^c. Tc = 1.54 K sample:
diamond, vecH _|_ ^c. ×, vecH ||
^c.
The fits in Figure are all to the simple
linear form
where t is the reduced temperature, T/ Tc. The
least-squares-determined values of Hc2(0) and
Tc 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 ith point sigmai 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 C4KHg Hc2(T) data. The individual
specimens are labeled by the value of Tc obtained
from the zero-field sweeps. The other parameters, including
the ``fit'' Tc, 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 Hc2 in their region of
validity, so this result is not surprising, especially since
Iye and Tanuma[120] found
linear temperature dependence for their C4KHg
samples. One feature of Table
that merits some discussion is that the Tc
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
Tc is a fairly common occurrence in
superconductivity which is generally caused by curvature of
the Hc2(T) data in the region near
Tc.[186]
Hohenberg and Werthamer[111] found that Fermi-surface
anisotropy can cause positive curvature of the critical field
near Tc, but here negative curvature is needed in
order to explain the high extrapolated values of
Tc.
For the higher- Tc specimens, the Tc
numbers found from the vecH || ^c fits are higher than
those found from the vecH _|_ ^c fits. Because of the
discussion of type I superconductivity in the Section ,
one might wonder if the Hc2, || ^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
Tc. The temperature dependence data for the
Tc 1.5 K GIC's for vecH ||^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 Hc2(theta). Here the data were obtained
directly, using field sweeps at constant theta as a
function of temperature.
Figure: Critical fields with vecH ||
^c for Tc 1.5 K C4KHg samples.
Uparrow marks the value of Tc found using a
zero-field temperature sweep. a) (bullet), data for a
Tc = 1.53 K sample; (diamond), a linear fit
to the data with Hc2(0) = 89.7 Oe, Tc =
1.65 K and cal R = 6.25e-3; (.), a quadratic fit to
the data with Hc2(0) = 64.0 Oe, Tc =
1.55 K and cal R = 1.2e-2. b) (bullet), data
for a Tc = 1.54 K sample; (diamond), a
linear fit to the data with Hc2(0) = 85.8 Oe,
Tc = 1.62 K and cal R = 1.62e-3; (.), a
quadratic fit to the data with Hc2(0) = 62.8 Oe,
Tc = 1.51 K and cal R = 4.7e-2.
Figure shows that using a
quadratic function to fit the vecH || ^c data does
improve the agreement between the fit Tc 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 C4KHg is
still unresolved, since the fits to Hc2, || ^c are
inconclusive. As was mentioned in relation to Figure ,
in C8K both Hc(T) and Hc2,_|_
^c(T) show positive curvature,[141] so perhaps in C4KHg
it is not implausible for both critical fields to show a
linear temperature dependence.
Some lower- Tc samples, such as the Tc
= 0.95 K specimen whose data is shown in Figure
(and others whose data are not displayed) showed
Tc'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
vecH || ^c measurements were usually taken going down
in temperature, and those for vecH _|_ ^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 Tc obtained from the Hc2, ||
^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 Hc2(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
Hc2(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 Tc,
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 ( Tc, <=ft.
dHc2/dT |Tc) WHH fits to the
C4KHg 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 Hc2(T) data taken on a Tc = 1.54 K
sample. a) (bullet), data for vecH _|_ ^c. (.),
linear fit with Hc2(0) = 748 Oe, Tc =
1.52 K, and cal R = 6.9e-3. (circ), WHH fit
with Hc2(0) = 518 Oe, Tc = 1.53 K, and
cal R = 1.6e-2. b) (bullet), data with
vecH || ^c. (.), linear fit with Hc2(0) =
85.8 Oe, Tc = 1.62 K, and cal R = 1.6e-3.
(circ), WHH fit with Hc2(0) = 59.76 Oe,
Tc = 1.63 K, and cal R = 1.2e-2.
As might be expected, the Maki-deGennes equation gives approximately linear behavior near Tc. Therefore the linear and WHH fits are in good agreement just below Tc. 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
Hc2(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 Hc2 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
fracdh*dt = -1 at t = 1.
The conclusions drawn here are perfectly consistent with
those stated by Iye and Tanuma in their papers on
C4KHg.[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 C4KHg, 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
Hc2(T) behavior may not occur in the lower-
Tc samples. Considering that the
Hc2(theta) for the lower- Tc
GIC's were well-described by the simple AGL model, it seems
reasonable that the lower- Tc Hc2(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 Hc2(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 Hc2(theta) data. The resultant numbers
are displayed in Table .
Table: Comparison of the anisotropy
parameter epsilon as obtained from Hc2(T)
and Hc2(theta) fits. The Hc2(T)
epsilon numbers were obtained from the ratio of the
slopes. The Hc2(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 Hc2(T) is
consistently about 9, that determined from
Hc2(theta) ranges from 8 to 16. The reason
that the angular dependence's epsilon's are higher is
that in fitting Hc2(theta) it has been
assumed that the number measured for Hc2(0°)
is actually Hc, and that the real value of
Hc2(0°) is much lower. As discussed in Section
, the Hc2(theta) data cannot be fit
without making use of this assumption. In addition, the
presence of type I superconductivity is supported by the
Cv 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 Hc2(theta) data, without benefit of fits. One way of doing this is to plot Hc2(theta)/Hc2(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 Hc2(theta)/Hc2(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
C4KHg, where 1/epsilon ==
Hc2(90°)/Hc2(0°). Data are for
a Tc = 1.54 K C4KHg sample.
(circ), t = 0.29. (bullet), t = 0.55.
(×), t = 0.76. All Hc2(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 Hc2, || ^c or
an upward deviation from linearity in Hc2, _|_ ^c.
While it is certainly true that neither of these trends is
obvious in the Hc2(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 C4KHg is taken
seriously, then the data plotted in Figure
b) are Hc rather than Hc2, and so make
no statement about any hypothetical curvature of Hc2, ||
^c(T). Any positive curvature of Hc2, _|_ ^c
would be lost in the noise of the data at low reduced
temperatures, so that any kink of the size predicted by the
Hc2(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 Hc2(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 Hc2(theta) data is much less affected by experimental errors, the magnitudes of epsilon determined from the Hc2(theta) fits are thought to be more reliable.
Temperature-dependent anisotropy has been observed before in
graphite intercalation compounds, specifically in
C8KHg 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 C4KHg. C8KHg is the only GIC
for which Iye and Tanuma showed Hc2(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
C8KHg is demonstrated by a plot of
Hc2(theta)/Hc2(0°) versus
theta, just as in Figure .
All data from Iye and Tanuma, Ref. [240] on a Tc = 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
Hc2(T) in C8RbHg. Data are taken from
Iye and Tanuma, Ref. [120].
(circ), Hc2, _|_ ^c. (bullet),
Hc2, || ^c. Parameters for the line fits: for
vecH _|_ ^c, Hc2(0) = 3078 Oe,
Tc = 1.36 K, and cal R = 0.56; for
vecH || ^c, Hc2(0) = 89.0 Oe, Tc
= 1.37 K, and cal R = 3.02e-2. Zero-field
Tc 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 C4KHg) show
positive curvature of their upper critical fields with
respect to temperature (d
Hc22/dT2 > 0). The
largest curvature seems to occur in C8RbHg, 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, Fe0.05TaS2[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 Hc2(T). These references should not be understood as suggestions that positive curvature is observed in C4KHg, but rather as expressions of the philosophy that whatever phenomena are the cause of the positive curvature in C8RbHg are most likely also to cause the extended linearity seen in C4KHg. 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 Hc2(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 C4KHg, 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, Hc(T) in C8K 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 Hc.[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 Hc2(theta) fits without it. The Hc2(theta) fits also suggest that the anisotropy parameter epsilon is temperature-dependent. Both the temperature dependence of epsilon and the enhanced linearity of Hc2(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.
