When a full set of critical field data was desired on a
sample, its zero-field transition was first obtained to make
sure that the particular specimen was in fact
superconducting. Then the sample was cooled to the lowest
obtainable temperature, usually about 0.44 K, at which a
series of field sweeps was performed as a function of the
angle between the sample c-axis and the applied field. The
definition of this angle, hereinafter called theta, is
illustrated in Figure 
a. The samples were
mounted with their a-axis vertical, and the field of the
magnet was parallel to the ground, so that the rotation was
accomplished by turning the probe by hand about its vertical
axis. The relative amount of rotation was determined by
fixing a pointer to the probe and noting its angular
displacement with respect to a metal compass bolted onto the
4He cryostat. Orientation readings made in this
fashion were thought to be reproducible to about ±
1°, an estimate based on the scatter in the
Hc2(theta) curves. The exact direction
corresponding to the samples' c-axis could not be determined
a priori and so was found by a fit to the angular
dependence after the data were entered into a computer at a
later date. The approximate orientation of the _|_ ^c and ||
^c directions was estimated at runtime by visual inspection
of the data, and this information was used for the
measurement of the temperature dependence of Hc2
at constant orientation.
After a complete angular dependence of Hc2 had
been measured at the lowest obtainable temperature, the
sample was rotated until the applied field was aligned
parallel to the carbon (graphene) planes. This alignment was
not perfect for several reasons. The first reason is the
inaccuracy in the theta reading. Secondly, the
specimens' a-axis may not have been exactly vertical due to
small inaccuracy in sample mounting. The misalignment angle
between the carbon (graphene) planes and the vertical will be
called ø to distinguish it from theta, the
angle in a horizontal plane between the graphite c-axis and
the applied magnetic field. (The angle ø is defined
pictorially in Figure .) Misalignment due to
this cause is estimated to be at most 5°, and probably
less. A third cause of imperfect alignment is that the carbon
(graphene) planes of the C4KHg specimens used in
these measurements are not exactly flat due to imperfect
alignment of the c-axis of the graphite host. The amount of
misalignment of the c-axis is called the mosaic spread. Also,
a small amount of exfoliation (non-uniform spreading of the
graphite layers) tends to occur during intercalation. Rocking
curves taken using elastic neutron (00l) scans (see
Chapter ) showed the mosaic
spread of a few C4KHg HOPG samples to be 2° to
3°. Some of the HOPG-based GIC's likely had larger mosaic
spreads. Mosaic spreads as large as 8 or 9 degrees are not
uncommon after intercalation in ternary and metal-chloride
compounds.[] The flattest
samples from a given batch were always chosen for the
critical field measurements, but nonetheless effects due to
sample warping could not be completely eliminated. Overall
inaccuracy due to all these causes could perhaps have been as
much as 7°, and is estimated to have averaged about
3° in practice. Because these alignment and flatness
problems tend to reduce the observed value of
Hc2_|_^c, the anisotropy values quoted in this
work should be thought of as representing lower bounds.
With the applied field parallel to the carbon (graphene) planes, a series of field sweeps at increasing temperatures was taken to determine Hc2_|_^c(T). The temperature was stabilized during these sweeps by rapid adjustment of the various valves in the pumping system. The temperature range in which the sample could be stabilized during the time necessary to complete the field sweep was about ± 3%, according to the thermometer. Of course, the GIC's temperature varied more slowly than the thermometer's due to its insulation from the bath by its encapsulation tube.
When the temperature series with vecH _|_ ^c was completed, a similar series with vecH || ^c was similarly performed. These measurements were much less sensitive to misalignment, but were susceptible (because of the smaller size of Hc2|| ^c) to electrical noise that generated stray fields. When the two temperature series were finished, two more sets of Hc2(theta) at constant temperature data could be collected: one at 1.2 K, which corresponds to the lowest obtainable temperature of the 4He bath with a roughing pump; and one set at 0.9 K, which corresponds to the lowest 4He temperature obtainable by using a booster pump in addition. Reliable Hc2(theta) data sets could not be obtained at other temperatures because of the impossibility of stabilizing the temperature for the necessary period of time (2 to 3 hours) without an electronic feedback system.
When a full set of data had been obtained, it was analyzed
graphically. Hc2 was defined as the intersection
of a line drawn tangent to the transition with the level
upper portion of the sweep, the same definition as was used
in Ref. [120]. The
application of this criterion for Hc2 is
illustrated in Figure b. Other definitions of
Hc2 were tried in analyzing the data; while they
slightly changed the results quantitatively, they had no
effect on the shape of any of the curves described
here. (See Figure for a demonstration of
this.) The possibility of bias introduced in the data
analysis is discussed further in Section .
The data were then typed in by hand to a VAX 11/750 computer
and fit by the appropriate formulae.
Electrical noise was not a problem in the critical field
determination since the signal-to-noise ratio for the field
sweeps was about 1000. However, the width and shape of the
transitions changed drastically as the angle theta was
varied. This lack of constancy in the shape of the transition
made a consistent definition of the upper critical field a
tricky matter. The different shapes of the transition are
illustrated in Figure . The change in breadth
of the superconducting transition with angle is common to
polycrystalline layered superconductors[200], and is due to the
contribution of misaligned grains as theta is varied.
Misaligned grains have almost no effect on the field sweep
when vecH is applied parallel to the c-axis, but will
contribute noticeably when vecH is applied in the
layer planes, perpendicular to the c-axis. The reason for
this behavior comes from the form of
Hc2(theta), which is strongly peaked near
vecH _|_ ^c. In essence, near the || ^c orientation,
since Hc2(theta) is fairly flat, slightly
misoriented grains have almost the same critical field as the
bulk of the sample. On the other hand, near the _|_ ^c
orientation, where Hc2(theta) is strongly
angle-dependent, slightly misoriented grains have much lower
critical fields than the bulk, and thus contribute to the
foot of the transition, making it appear much broader than it
would in a perfect crystal.
To some extent, the greater breadth of the vecH _|_ ^c transition may be an intrinsic effect. The reason is that the anisotropy of Hc1 in a superconductor described by the anisotropic Ginzburg-Landau theory is expected to be approximately the reciprocal of the Hc2 anisotropy.[155] That is, theoretically Hc1 should be lower in the direction where Hc2 is higher, so that the transition should be broader in the high- Hc2 direction. It is not clear whether the Hc1 anisotropy is also contributing to the orientation dependence of the transition width.
Figure: a) Definition of the angle
theta, the angle between the applied magnetic field
and the graphite c-axis. This angle is the complement to that
usually used in the thin-film superconductivity literature,
but corresponds to customary usage in the GIC literature. b)
A sketch showing how Hc2 is determined graphically
from raw susceptibility versus magnetic field data. Note the
similarity of this trace to Figure
b).
Figure: a) Superconducting transitions with
the magnetic field applied parallel and perpendicular to the
graphite c-axis for a typical C4KHg sample. Notice
how much broader the transition is in the vecH_|_ ^c
case. b) Similar data from Iye and Tanuma, Ref.[120], Figure 2.
There are several ways in which the critical field experiments described in this chapter could have been improved. Firstly, a servo-controlled gearing system for rotating the sample would probably reduce the errors in the Hc2(theta) measurements, and would certainly make these measurements easier to perform. Probes were available that have gearing systems for rotating the sample around a horizontal axis, but not for rotating the sample about a vertical axis, as was necessary in this experiment. Secondly, an electronic temperature controller would have made it possible to obtain Hc2(theta) scans at much more closely spaced intervals below Tc, and thus would have allowed taking many points in an anisotropy versus temperature curve. Thirdly, performing the data collection with a computer rather than a chart recorder would greatly reduce the labor involved in reducing the data, and also would allow more sophisticated real-time data analysis. Lastly, if a more powerful x-ray apparatus were available for doing diffraction on samples in glass tubes, it would be possible to measure the mosaic spread of the crystals directly. This measurement would eliminate some of the uncertainty in the fitting of Hc2(theta) described in the next section. All in all, it is expected that improvement of the instrumentation would have a real impact on the Hc2(T) experiment, but would not affect the Hc2(theta) data much because of the limitations due to crystalline quality and misalignment.