As was discussed in Section ,
one of the possible extrinsic causes of the deviations in
H_{c2}(*theta*) is that the GIC's a-axis may not
have been mounted perfectly vertically. Figure
a) shows the properly aligned case for reference, and Figure
b) illustrates the case where the sample is placed
inside the inductance bridge with its c-axis rotated by an
angle ø from the horizontal. In both a) and b), the
laboratory vertical is called ^z. The sample in-plane axes
are denoted by ^a and ^a^{'}. Thus in a) ^z and
^a^{'} are parallel, whereas in b), the misaligned
case, the angle between ^z and ^a^{'} is also
ø. Since the measurements described here were done on
HOPG, the sample's c-axis is a well-defined crystallographic
direction, but the crystallites are randomly oriented
in-plane, and so the directions ^a and ^a^{'} are
designated merely for convenience.

The important point to notice here is that in b) the ^a axis
is still in the horizontal plane, even though the c-axis is
not. Therefore rotation of the sample about ^z will allow
correct measurement of H_{c2,|| ^a} = H_{c2,_|_
^c}, but the value measured for H_{c2, || ^c}
will be off be a factor corresponding to ø degrees. In
fact, the minimum in the measured
H_{c2}(*theta*) will come when *vec*H is in
the plane defined by the vector ^c and ^z. Through use of
Eqn. , one can see that the
minimum measured critical field for the misaligned case will
be H_{c2}(0°)/*sqrt**cos*^{2}
ø + *epsilon*^{2} *sin*^{2}
ø. With a bit more effort, by following the original
derivation of Morris, Coleman, and Bhandari[175] the formula for
H_{c2}(*theta*) for a specimen tilted by an
angle ø can be derived:

The detailed derivation of this formula is outlined in
Appendix . If one lets H_{c2,||
^c}^{eff} == H_{c2,||
^c}/*sqrt**cos*^{2}ø +
*epsilon*^{2}*sin*^{2}ø and
*epsilon*^{eff} ==
*epsilon*/*sqrt**cos*^{2}ø +
*epsilon*^{2}*sin*^{2}ø,
then one recovers the form only with the effective
quantities replacing the actual ones. The impact of a
non-zero tilt angle is therefore to increase the measured
value of H_{c2,|| ^c} but not to change the measured
value of H_{c2,_|_ ^c}, therefore reducing the
apparent anisotropy, 1/*epsilon* =
H_{c2,_|_^c}/H_{c2,|| ^c}. Since the
experimental data in Figure
b) is higher than the fit near *vec*H||^c, sample tilt
seems like a good candidate for explaining the discrepancy.
However, the impact of tilt for a plausible range of ø
is quite small since H_{c2}(*theta*) is quite
flat near *theta* = 0. The minute amount of distortion
of the curves that occurs for a believable tilt angle
(ø <= 5°) is demonstrated in Figure .
After examining this figure one can conclude that the effect
of specimen tilt on H_{c2}(*theta*) is probably
unimportant.

**Figure:** Why a tilted sample affects the
shape of H_{c2}(*theta*). The notation (
*vec*H *cdot* ^x)^x signifies the projection of
*vec*H along ^x. a) The aligned case. Rotations of the
sample around the vertical ^z allow *theta* to be varied
all the way from ( *vec*H||^c) to ( *vec*H_|_^c) (
*vec*H_|_^c = *vec*H|| ^a). b) The misaligned case.
H_{c2,_|_^c} can still be measured correctly, but
instead of the true value of H_{c2||^c} one will get
H_{c2,||^c}/*sqrt**cos*^{2}ø
+
*epsilon*^{2}*sin*^{2}ø.

**Figure:** The effect of sample tilt on
H_{c2}(*theta*). The three curves in this
picture were calculated using the parameters H_{c2,||
^c} = 42 Oe, anisotropy == 1/*epsilon* = 15 and the
following values for the tilt angle: (*circ*) ø =
0°; (*bullet*) ø = 10°; (*diamond*)
ø = 40°. The ø = 0° curve corresponds
to one of the fits shown in Figure
b). Note that the curves for ø = 10° and for
ø = 0° are almost indistinguishable.

Besides macroscopic tilt of the specimen, there is another
type of misalignment that can influence the shape of the
H_{c2}(*theta*) curves. A microscopic type of
misalignment is mosaic spread, which in this context means
the half-width-at-half-maximum-intensity (HWHM) of a GIC's
(00*l*) peaks in a diffractometer rocking scan. The
mosaic spread is a measure of the degree to which the c-axes
of the crystallites in a piece of HOPG are aligned. Clearly
if the planes of a GIC are not flat, the peak in
H_{c2}(*theta*) will be somewhat smeared out.
Therefore, a finite amount of mosaic spread will reduce the
anisotropy ratio by reducing H_{c2}(90°) =
H_{c2,_|_ ^c}. The effect of mosaic spread on the
critical fields of the GIC superconductors was estimated by
convolving the formula in Equation
with a gaussian function to represent the probability of
misalignment as a function of misalignment angle. With a
gaussian distribution of misaligned crystallites, for each
increment *Delta* away from perfect alignment, there is
a reduction by a factor 1/**e** in probability. Therefore
only orientations within about *Delta* degrees of
perfect alignment contribute significantly. The form of the
expression is:

where *theta* is the angle for which the critical field
is being calculated, *theta*_{i} is the dummy
variable (which runs from 0° to 180°), and
*Delta* is the mosaic spread in degrees. The results of
the calculations are exemplified by the curves in Figure
.

**Figure:** The effect of mosaic spread on
H_{c2}(*theta*), calculated using Equation
. The same parameters were used as in Figure ,
except that here the mosaic spread, *Delta*, is varied:
(*circ*) *Delta* = 0°; (*bullet*)
*Delta* = 3°; (*diamond*) *Delta* =
10°.

The most appropriate HWHM of the gaussian would be the mosaic
spread measured by (00*l*) diffraction, if known. In the
present work the mosaic spread was unknown, and so
*Delta* was used as a parameter in the fits. Because a
non-zero mosaic spread makes the H_{c2}(*theta*)
curves broader, it actually worsened agreement between
experiment and theory, since the theoretical curves are
already too broad in the wings of the peak. This finding does
not, of course, indicate that the GIC's used in these
experiments were perfect crystals; rather it is an indication
that the real cause of the unsatisfactory fits for the
T_{c} = 1.5 K samples has not yet been identified.

The presence of multiple crystalline or disordered phases in
C_{4}KHg is another property that could impact upon
the angular dependence curves. The differing properties of
the two phases are discussed in detail in Chapter .
As far as the H_{c2}(*theta*) curves go, it
suffices to say that since the phases may have different
T_{c}'s, they could have different upper critical
fields. If these two phases are distributed inhomogenenously
in the GIC's, then at the lowest temperatures, where both of
them are superconducting, their presence could distort
H_{c2}(*theta*). However, the temperature and
sample dependence of the quality of the fits weigh against
this interpretation. For one thing, Figure
b) shows that the H_{c2}(*theta*) curves are
quite reproducible from sample to sample, an unlikely
occurrence if the distribution of phases is important.
Secondly, as mentioned in connection with Figure ,
the quality of the fits improves with decreasing temperature,
the opposite of what would be expected if a lower-
T_{c} minority phase were causing the deviations.
Thirdly, the samples which show larger deviations from the
AGL functional form have narrower zero-field transitions than
the lower- T_{c} specimens which are well-fit by Eqn.
. X-ray and neutron diffraction experiments confirm that
the higher- T_{c} samples are actually more uniform
than the lower- T_{c} ones.

The last of the extrinsic factors that needs to be considered
as a explanation of the disagreement between the
H_{c2}(*theta*) data and fits to the data is the
possibility of bias influenced by the method of data
reduction. As was discussed in Section ,
the standard working definition of H_{c2} (for
inductive transitions) was used; this procedure gives
H_{c2} as the intersection of a tangent drawn to the
most linear part of the field sweep with the level upper
portion of the trace. The unavoidable question when
encountering difficulties in fitting H_{c2} data is
whether there is any reason to expect agreement given that
the theoretical definition of H_{c2} is so different
from the working one. That is, there is no guarantee that the
field found by the procedure described above corresponds to
the highest field where vortices can nucleate in the
superconductor, which is the theoretical definition of
H_{c2}.[252,108] The possibility of bias
due to analysis is particularly troublesome for the study of
H_{c2} in anisotropic superconductors, where the
transition shape can be a strong function of *theta*, as
Figure b) shows.

The only response that an experimentalist can make to such
criticism is that the procedure defined above is as good as
any available, and that it is capable of producing data which
are well-fit by Equation , as Figure
a) shows. In order to check for any bias introduced by the
tangent method of critical field determination used here, the
alternative definition of H_{c2} as the 90%
completion point of the transition was also tried in
analyzing some H_{c2}(*theta*) data sets. Figure
is a comparison of the
curves obtained using the two different analysis techniques.
As this juxtaposition shows, slightly different curves are
produced by the two analysis methods. However, the agreement
with Eqn. is not improved.
Therefore it is thought that the unsatisfactory quality of
the fits to the T_{c} = 1.5 K samples is not an
artifact of the analysis. In a similar vein, Decroux and
Fischer have noted that H_{c2} results on the
molybdenum chalcogenide compounds (also called Chevrel
phases) are not biased by the critical field definition as
long as the compounds do not contain magnetic ions.[53]

**Figure:** Comparison of the effect of the two
definitions of H_{c2} on
H_{c2}(*theta*). *circ*, tangent
definition; *diamond*, 90% definition. Data are for a
T_{c} = 1.53 K sample at T/T _{c} = 0.29. The
90% method tends to produce slightly higher critical fields,
but there is only a small difference between the shapes of
the curves for the two analysis methods. Use of the 90%
definition does not improve agreement with Eqn.
.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995