The three factors
described above, namely sample tilt, mosaic spread, and
critical field definition, are the extrinsic influences which
may affect the shape of H_{c2}(*theta*). None of
these is thought to have a large effect on the data. There
are also more fundamental, intrinsic explanations for the
disparity between data and fit. Probably the most applicable
of these to C_{4}KHg is the conceivable existence of
type I superconductivity for field orientations near
*theta* = 0°. Type I superconductivity for 0°
<= *theta* <= 25° has been previously reported
for C_{8}K by several groups[137,136,141]. C_{9.4}K[137,136] and C_{8}Rb[133], on the other hand,
are found to be type I for all field directions. The
existence of type I character for a range of angles in
layered superconductors was predicted even before its
experimental discovery by Kats.[127] For these reasons the question
of type I superconductivity in C_{4}KHg should be
approached with an open mind, despite the type II
superconductivity reported for all field orientations in Ref.
[120].

**Figure:** H_{c2}(*theta*) for
C_{8}K, from Ref. [141]. The fields are labeled
H_{c2} in the type II region and H_{c3} and
H_{c} in the type I region.

Iye and Tanuma concluded that there is no type I
superconductivity in C_{4}KHg because they did not
observe supercooling in their field sweeps for any angle
*theta*.[120] However,
the presence of supercooling and the differential
paramagnetic effect (DPE) are not the only criteria for the
determination of the type of superconductivity. Another
approach to the question is to look at the Ginzburg-Landau
*kappa* parameter since *kappa* = 1/*sqrt*2 is
the critical value dividing type I and type II
superconductors. *kappa* can be estimated using specific
heat and critical field data through use of the following
formulae:[252]

where *Lambda* is the magnetic field penetration depth,
*xi* is the coherence length, H_{c}(0) is the
zero-temperature thermodynamic critical field, and
*Gamma* is the normal-state linear electronic specific
heat coefficient. The equality between *kappa* and
*kappa*_{1} is good to within about 20%,[69] so we can use
*kappa*_{1} as a rough esimate of *kappa*.
(In the discussion below, what is referred to as *kappa*
has actually been calculated using the definition of
*kappa*_{1}.) Because *Gamma* has been
measured for C_{4}KHg,[8] one can estimate
*kappa*(0) by calculating H_{c}(0) and linearly
extrapolating H_{c2}(T) to zero temperature. The
applicability of this procedure to the T_{c} = 1.5 K
GIC's is uncertain because no superconducting transition was
observed down to 0.8 K in the samples on which *Gamma*
was measured. It seems possible that *Gamma* could be
different in the T_{c} = 1.5 K and T_{c} =
0.7 K samples, even though it is a property of the normal
state. Also, *Gamma* was given in Ref. [8] in molar units, so in order
to convert to cgs units one must calculate the molar volume,
which means that one must assume a structure for the
compound. Here a (2 × 2)R0° in-plane unit cell is
employed, since this is the structure which has most
frequently been observed in C_{4}KHg.[123,246] (See Figure
.) Despite these caveats, since no other measurement of the
specific heat is available, it is useful to see what can be
learned by estimating *kappa*. The results of this
calculation are given in Table
for three MIT C_{4}KHg samples and three from Ref.
[120]. Notice that the
magnitude of *kappa* is generally lower for the higher-
T_{c} specimens. The implication is that type I
transitions are more likely to be observed in the higher-
T_{c} GIC's.

**Table:** Calculated values of *kappa*
for selected C_{4}KHg samples. The H_{c2}(0)
numbers come from linear extrapolation of the
H_{c2}(T) data (see Section ),
and so are independent of the H_{c2}(*theta*)
measurements.

The meaning of *kappa* will be discussed in more detail
below, but for the moment it suffices to say that
*kappa*_{|| ^c} tend to be quite close to
*kappa*_{critical} = 0.707. The primary
conclusion from this calculation is that within the
uncertainty of the approximations made, the type of the
superconductivity in C_{4}KHg is indeterminate, but
that it could very well be type I for some samples for
*theta* close to zero. This is especially true at higher
temperatures since, according to the two-fluid model:[252]

where **t** is the reduced temperature. Thus it seems
quite likely that the presence of type I superconductivity is
a factor in explaining the poor fits obtained to
H_{c2}(*theta*) data with Equation ,
at least in some specimens at higher temperatures.

Since *kappa*_{|| ^c} decreases with increasing
temperature, the angular range in which type I behavior
occurs should increase as t increases. The variation of the
angular range of type I character is observed in TaN, the
only non-GIC bulk material known to display type I or type II
character depending on field orientation.[259] Some
H_{c2}(*theta*) at constant temperature curves
for TaN are shown in Figure . In the bottom trace, at
T = 1.65 K, TaN is type II for all angles, as the excellent
agreement with the dashed fit indicates. When the temperature
is increased to 1.8 K, the sample has type I character near
the [100] direction, with the result that the data there
deviate above the dashed curve. When the temperature increase
to 2.1 K, the measured critical field for almost all angles
*theta* is H_{c}, and when the temperature
reaches 2.4 K, the sample is type I for all orientations.
C_{8}K might also be type I for all orientations for
temperatures sufficiently close to T_{c}. The change
with temperature of the H_{c2}(*theta*) fits for
C_{4}KHg is qualitatively similar, although less
dramatic, since type II behavior persists for a much larger
range of angles. For the higher- T_{c}
C_{4}KHg specimens, the H_{c2}(*theta*)
curves are probably most comparable to the 1.8 K TaN trace in
Figure
.

**Figure:** H_{c2}(*theta*) curves
for TaN showing a transition from type I to type II character
as a function of field direction. From Ref. [259]. TaN is the only bulk
superconductor besides C_{8}K (and possibly
C_{4}KHg) known to display this variability. The
temperatures at which the curves were taken and the
thermodynamic critical fields are indicated. Note that at
1.65 K, the sample is entirely type II, but that at 2.4 K it
is entirely type I.

If the angular range of type I behavior increases with
temperature, then the agreement between Eqn.
and the C_{4}KHg data should deteriorate as the
temperature increases. The validity of this assertion is
demonstrated by Figure , where the fit clearly
worsens as the temperature approaches T_{c}.

**Figure:** Anisotropic Ginzburg-Landau model
fits (dotted curves) to H_{c2}(*theta*) data as
a function of temperature. All fits were produced with the
parameters tilt = 0° and mosaic spread = 0°. a) t =
0.29, H_{c2,||^c} = 47 Oe, anisotropy
(1/*epsilon*) = 9.5, and residual parameter *cal* R
= 1.18. b) t = 0.57, H_{c2, ||^c} = 33 Oe, anisotropy
(1/*epsilon*) = 5.5, and *cal* R = 1.25. c) t =
0.78, H_{c2,||^c} = 23.1 Oe, anisotropy
(1/*epsilon*) = 4.5, and *cal* R = 1.43.

If type I superconductivity is indeed present in
C_{4}KHg, this fact should be accounted for in
fitting the data. In Figure , new fits are shown in
which the theoretical curve has the form:

The idea behind using this functional form is that if
H_{c} is greater than H_{c2}, then
H_{c} will presumably be measured as the critical
field. A quick comparison of Figs.
and
shows that inclusion of H_{c} as a parameter
dramatically improves the quality of the fits at all
temperatures. This large improvement is in contrast to the
small (or negative) impact on the quality of fit of the
mosaic spread and tilt parameters. Residuals for the fits
with and without type I behavior are gathered in Table
. The improvement in the residual index is not accounted
for simply by the addition of another free parameter since
the number of parameters is taken into account in the
definition of *cal* R (see Eqn. ).

**Figure:** Anisotropic Ginzburg-Landau model
fits (dotted curves) to H_{c2}(*theta*) data as
a function of temperature, taking into account the
possibility of type I behavior. All fits were produced with
the parameters tilt = 0° and mosaic spread = 0°. a) t
= 0.29, H_{c2,||^c} = 35 Oe, anisotropy
(1/*epsilon*) = 14, H_{c} = 65 Oe, and residual
parameter *cal* R = 0.39. b) t = 0.57, H_{c2,
||^c} = 19 Oe, anisotropy (1/*epsilon*) = 15.5,
H_{c} = 43 Oe, and *cal* R = 0.84. c) t = 0.78,
H_{c2,||^c} = 14.5 Oe, anisotropy (1/ *epsilon*)
= 12.5, H_{c} = 24.5 Oe, and *cal* R =
1.11.

For these new fits, H_{c} was taken as a free
parameter. The numbers obtained for the thermodynamic
critical field from the H_{c2}(*theta*) fits for
two samples are shown in Table ,
and H_{c}(T) is plotted in Figure .
According to the BCS theory, H_{c}(T) should be
quadratic, having the form (1 - t^{2}). The fact that
the H_{c} values found from the angular dependence
are better fit by a straight line is somewhat disturbing,
since (1 - t) behavior is expected for H_{c2}(T) (see
Section ). However,
H_{c}(T) showed *positive curvature* in
C_{8}K (see Figure ), where signs of type I
superconductivity (large hysteresis and differential
paramagnetic effect) were unmistakable.[141] Since the temperature
dependence of H_{c2}(T) is unusual in
C_{4}KHg, perhaps it is not straining credibility too
much to suggest that H_{c}(T) could be anomalous,
too, just as in C_{8}K.

The obvious question is whether these values of the critical
field are consistent with those calculated from the specific
heat measurements, also given in the table. The calculated
thermodynamic critical field tends to be about 1.8 times the
fit one. The most likely explanation for this inconsistency
is that the value of *Gamma* quoted in Ref. [8] is simply too high. For the
reasons mentioned above, a higher value of *Gamma* for
the sample of Ref. [8]
would not be very surprising.

Another possible explanation for the difference in magnitude
of H_{c} is that up to this point the demagnetization
factor *cal* D has not been taken into account. (1 -
*cal* D) H_{c} is the field where normal regions
are first formed.[252,157] Thus, the ideal type
of magnetization curve shown in Figure a
is found only for ideal samples with *cal* D = 0; in
real specimens the sharp corner is rounded. The number, size,
and shape of the normal regions formed for (1 - *cal* D)
H_{c} ;SPM_{l}t; H_{applied}
;SPM_{l}t; H_{c} depends typically on sample
size, shape, defects, and orientation with respect to the
applied field.[157]
The primary effect of a non-zero demagnetization factor then
is to change the shape of the transition with angle. Because
of the method of critical field determination used here (see
Figure b), the possible
influence of demagnetization on the transition shape cannot
be completely ruled out. Therefore, demagnetization-induced
distortion could be causing the measured critical field to be
substantially less than the actual value of H_{c}.
The influence of demagnetization effects should be
particularly strong for the conditions used in the study of
C_{4}KHg, since for thin plates in a transverse field
(here *vec*H || ^c) *cal* D 1.0, meaning that the
samples are effectively always in the intermediate state.[157]

**Table:** H_{c}(t) values for
C_{4}KHg obtained from fits to Eqn. .
For comparison, H_{c}(t) calculated from the specific
heat data for a T_{c} = 1.53 K sample is also
included. The samples' H_{c2}(*theta*) data are
shown in Figs. , ,
and
.

Ideally it would be possible to obtain the demagnetization
factor as a function of angle in order to correct the
H_{c2}(*theta*) and H_{c}(T) curves.
Past experimentalists have gotten a value for *cal* D by
machining a material with known critical fields to the size
and shape of the sample.[59] Then one gets *cal* D by
comparing the measured critical fields of the machined
specimen to the known values for shapes with zero
demagnetization. Unfortunately this procedure would have to
be carried out for each specimen. If a reliable calculation
of *cal* D were available, it would also allow a more
rigorous testing of the theory. However, since the
demagnetization is strongly dependent on the sample
dimensions[59] and
because *cal* D could easily be affected by such
hard-to-quantify factors as exfoliation, the amount of effort
required does not seem worthwhile. Any use of a value for the
demagnetization would also require the introduction of
several new parameters.

An ideal type I superconductor should be easy to distinguish
from a type II superconductor because of the discontinuity in
its magnetization at <=ft| *vec*H *|* =
H_{c}. (See Figure ).
However, a non-zero demagnetization factor can smear out the
magnetization curve, making even type I transitions appear
more or less continuous. Despite the presence of
demagnetization effects, it seems sensible to examine the
field sweeps to see if there is any evidence of discontinuity
under the conditions where type I behavior is suspected.

In the lower- T_{c} C_{4}KHg samples (whose
H_{c2}(*theta*) curves were well-fit by Eqn.
), the experimental traces appear smooth for all angles, as
Figure a) shows. The only major
difference between *theta* = 90° and *theta* =
0° is that the high-angle transitions are considerably
broader, for reasons discussed in Section .
This result is consistent with the finding that these samples
were type II for all orientations, and is consistent with Iye
and Tanuma's findings.[120]

In the T_{c} = 1.5 K samples, the field sweeps for
*vec*H _|_ ^c look smooth, just as they do for the
lower- T_{c} GIC's. For *vec*H || ^c, on the
other hand, the curves appear fairly continuous on the
upsweep, but develop a corner near the upper critical field
on the downsweep which could be identified as a
discontinuity. The arrow in Figure
b) indicates the location of the corner. While it is tempting
to identify this corner as a discontinuity stemming from a
first-order transition, it is a fairly small feature.
Therefore, the evidence regarding type I versus type II
behavior from the field sweeps is suggestive but ambiguous.
Low-temperature magnetization measurements would probably be
easier to interpret than susceptibility in this regard, but
unfortunately there was no readily available magnetometer
that could be cooled to the necessary 1 K range.

**Figure:** Comparison of field sweeps between
type II and possible type I transitions. The vertical
direction is the inductive voltage, while the horizontal
direction is field. All traces taken at about 0.4 K. a)
Transitions with *vec*H _|_ ^c and *vec*H || ^c for
a T_{c} = 0.95 K sample. For both orientations the
transitions appear smooth. b) Transitions with *vec*H
_|_ ^c and *vec*H || ^c for a T_{c} = 1.5 K
sample. For *vec*H _|_ ^c, the transition looks smooth,
consistent with its expected type II character. For
*vec*H || ^c, on the other hand, there is a small
discontinuity in the susceptibility near the upper critical
field which is indicated by an arrow. This feature was seen
consistently in T_{c} = 1.5 K samples.

**Figure:** Thermodynamic critical fields
obtained from H_{c2}(*theta*) fits versus
temperature for T_{c} = 1.5 K C_{4}KHg
specimens. The numbers plotted here are the same as in Table
.
(*bigtriangleup*), data for a T_{c} = 1.53 K
sample; (*circ*), data for a T_{c} = 1.54 K
sample; (*diamond*), a linear fit to the data with
H_{c}(0) = 85.2 g; (*circ*), a quadratic fit to
the data with H_{c}(0) = 66.5 g; (×),
H_{c}(t) calculated using the specific heat data of
Alexander *et al.*,[8] which gives H_{c}(0)
= 112 Oe.

Why should the lower- T_{c} specimens be type II for
all orientations, and the higher- T_{c} ones be type
I for some orientations? The superficial answer is that
H_{c2} isn't as strongly T_{c}-dependent as
might be expected, since the critical fields of the
T_{c} = 1.5 K samples are only slightly higher than
those of the T_{c} = 0.7 K samples. (Refer to Table
for the relevant numbers.) If H_{c} increases as
T_{c}^{2} and H_{c2} increases only
weakly with T_{c}, the net result must be a decrease
in *kappa* with increasing T_{c}. The deeper
answer to the question is that *xi* and *Lambda*,
the lengths whose ratio determines *kappa*, are both
affected by changes in v_{F} and *l*, the Fermi
velocity and the mean-free-path.[186] Therefore, one can invoke
differences between the two sample types either in the Fermi
surface or in crystalline perfection to explain dissimilarity
of their critical fields. Of course, the same mechanism
should ideally also explain their different zero-field
transition temperatures. Since the lower- T_{c} gold
specimens have broader zero-field superconducting transitions
than the higher- T_{c} pink samples (see Table
), it seems natural to think that the gold samples are less
well-ordered. This line of reasoning implies that the gold
specimens have a lower mean-free-path. The differences
between the gold and pink samples are discussed at length in
Chapter . In the meantime, the
implications of possible type I superconductivity in
C_{4}KHg deserve further consideration.

Another good question about the H_{c2}(*theta*)
curves is why the data are not perfectly flat near
*theta* = 0 if the measured critical field is
H_{c} there. Some angular dependence of H_{c}
is also present in the TaN data of Figure . As was
first suggested by P. Tedrow[242], the angular dependence of
the demagnetization factor could influence the shape of the
field sweeps enough to cause a perceived angular dependence
of H_{c}. Koike *et al.*[141] in their study of
C_{8}K also noticed a small angular dependence of
H_{c} near *theta* = 0 (see Figure
). In addition, they measured a strongly angle-dependent
supercooling field which they thought must be H_{c3},
the surface nucleation field. However, H_{c3} is not
angle-dependent in the sense indicated in Figure ;
it is in fact equal to 1.69 H_{c2}(theta), but here
*theta* is the angle between the crystal surface and the
c-axis, not the angle between the applied field and the
c-axis.[187] Thus
H_{c3}(*theta*) can only be measured by
preparing different crystals with differently oriented
surfaces, not by turning the applied magnetic field.

There is another problem with the identification of the
angle-dependent supercoooling field with H_{c3}. The
problem is that H_{c3} is the surface nucleation
field for a field applied *parallel* to the
surface,[54], while
*vec*H || ^c corresponds to the field
*perpendicular* to the surface. In some anisotropic
Chevrel phases with an unusual grain structure,
H_{c3} has been seen for the ``wrong'' orientation,
but for HOPG-based GIC's such as the ones studied by Koike
and coworkers, no such microstructural anomalies are
anticipated.[187]
Therefore the identification of the supercooling field || ^c
with H_{c3} must be mistaken. It seems likely that
the angular dependence of the supercooling field reported in
Ref. [141] is a
demagnetization-derived effect, along with the slight angular
dependence of H_{c} observed in C_{8}K[141] and C_{8}Rb[134]. Certainly the
change in the demagnetization *cal* D (from 1 for vecH
perpendicular to a plate to 0 for *vec*H parallel to a
plate) is large enough to cause this apparent angular
dependence.[242,252]

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995