The superconducting transitions were measured by monitoring
the ac susceptibility of the sample as the temperature or
magnetic field was swept. An ac inductance bridge was used to
record the sample's susceptibility. The homemade inductance
bridge, which is sketched in Figure 
,
was composed of a primary solenoid, around which two
counterwound secondary solenoids about half the length of the
primary were placed. The secondaries were balanced to within
about 2 microvolts under the typical operating conditions of
the experiment. The off-balance voltage included
contributions from the sample holder and small unavoidable
differences between the secondaries. This voltage was nulled
by using the zero adjust knob on the lock-in amplifier (see
Figure ) before the field and
temperature sweeps began. Several different versions of the
inductance bridge were used in the early zero-field
experiments, but all critical field measurements reported
here were done with the same unit.
Figure: A schematic drawing of the
inductance bridge. The sample capsule was placed inside the
primary coil. The windings were made from 38 gauge magnet
wire. There were 20 complete layers of winding in the
secondary coils and 2 complete layers of winding in the
primary.
Figure: A schematic of the data acquisition
system. For zero-field temperature sweeps, the thermometer
voltage was attached to the x-input of the flatbed plotter.
For fixed-temperature magnetic field sweeps, the dc output of
a stepping motor on the magnet power supply was attached to
the x-input of the plotter.
The fraction of a sample which is superconducting can be
estimated in the inductive measurements because the
transition height is proportional to the fraction of the
primary coil which is filled with superconducting material.
The exact expression, from Faraday's Law in cgs units,
is:
where Vsecondary is the superconducting transition height, A is the effective area of the superconducting material in the direction perpendicular to the primary modulation field, Bmod is the modulation field, and Omegamod is the modulation frequency. The dependence on the area perpendicular to the modulation field comes from the fact that this area determines the time-rate of change of magnetic flux in Faraday's Law. Here alpha is a dimensionless constant of the order of unity which takes into account the coupling of the primary coil to the secondary. This constant can in principle be calculated[2], but in practice it is preferable to simply measure alpha by filling the primary coil completely with a superconducting material, so that A = Acoil. Once alpha is determined in this way, a measurement of the transition height for a given specimen gives its superconducting cross-sectional area. From knowledge of the sample's total area, an estimate of the fraction which is superconducting (fraction = Asuperconducting/ Atotal) can be made, as discussed below.
It is important to note that this areal fraction is an upper
bound on the volume fraction of superconductivity.
To see why, consider the case where the modulation field is
applied along ^z, so that the contribution to the signal
comes from planes of constant z from z=0 to
z=L. [See Figure .] Portions of the sample
at all positions along the coil axis can contribute equally
to the effective area. For a given position (x,y) in the
cross-section, though, only one volume element can
contribute. This volume element will screen all others at the
same (x,y) position, so that A is really the area of the
projection along ^z of all the superconducting volume
elements onto a plane of constant z. Therefore the
cases where 100% of the volume superconducts and just one
full sheet (say that at z=L) superconducts cannot be
distinguished by this technique.
The inductance bridge used in the critical field measurements was calibrated by filling its end with aluminum foil. The foil had a Tc of 1.1 K (in surprisingly good agreement with tabulated values[10]) with a transition height of 38 muV. The thermal drift of the inductance bridge off-balance is about 50 nV/K, so it is estimated that the minimum observable transition height is about 100 nV. From the known area of the primary coil of 50.3 mm2, it was estimated that the transition height of a typical GIC should be about 5 muV, as was indeed the case. Thus the inductive experiment should be sensitive to superconductivity in about 2% of a typical sample. Unfortunately, repositioning a large GIC inside the primary coil changed its transition height by as much as a factor of 5. This variation occurs because the secondary coil is not much longer than the sample, and the secondary is much more sensitive to inductance changes near its center than near its end. A larger inductance bridge could not be accomodated inside the available 3He refrigerators.
Figure: Illustration of how a sample which
is only partially superconducting can mimic full
superconductivity in an inductive transition.
As a practical matter, the GIC's used in these experiments
are thought to be fairly homogeneous on a macroscopic scale.
Therefore, in general the effective areal fraction of
superconductivity should be approximately the volume one as
well. In the cases where the GIC's were multiphased, the
matter of superconducting volume and screening deserves more
thought, as discussed in Section .
The full data acquisition system is sketched in Figure
. The primary excitation current was supplied by an
oscillator, either one inside the lock-in amplifier, or an
external one attached to the lock-in's reference port. The
usual excitation current was 1 milliampere at 490 Hz. This
frequency was chosen because it seemed to be particularly
free of electrical noise. Neither the critical fields nor
Tc obtained from this apparatus were found to be
sensitive to either the magnitude or frequency of the primary
current. The primary magnetic field produced under this
excitation was calculated to be 0.17 oersteds (compared to
the 0.2 oersteds at 27 Hz used in Ref. [120]). These conditions were
sufficient to give a typical signal-to-noise ratio of about
1000, with the superconducting transition height usually
between 5 and 10 microvolts. The detailed theory of operation
of an inductance bridge circuit is described in Ref. [2].
The output signal of the inductance bridge was filtered by the lock-in, and the lock-in's output was fed to the y-input of a flatbed plotter. The x-input for temperature sweeps was either the voltage across a calibrated germanium resistor (Lake Shore GR-200A-100), or the dc output of a capacitive pressure sensor (MKS Baratron) which was attached to the 3He refrigerator. The temperature derived from the pressure sensor was in excellent agreement (the disagreement about the same as the interpolation error) with that obtained from the germanium resistor down to about 1.0 K, below which deviations increased with decreasing temperature. At the lowest temperatures (T 0.4 K), the germanium resistor consistently showed readings about 30% higher than those of the 3He vapor pressure under hard pumping, with the thermometer reaching agreement with the vapor pressure to within a few percent after about one hour's wait. The implication here is that a well-calibrated thermometer is essential for work below 1.0 K. At the lowest temperatures, the reading of the thermometer was always recorded in preference to the vapor-pressure-derived temperature. The magnetoresistance of the thermometer was negligible throughout the magnetic field range of interest, and was not corrected for.
Besides the question of agreement between the temperature sensors, there is also the problem of thermal equilibrium between the temperature sensors and the sample. The sample tube contained about 200 torr of helium gas intended to provide thermal contact of the GIC to the bath and to minimize any temperature lag. Furthermore, the sample was allowed to sit at constant temperature for about five minutes prior to each field sweep if the temperature had been changed. The actual degree of equilibration that was achieved with these precautions can be estimated by comparing the values of Tc obtained in heating and cooling temperature sweeps. In general, the offset between the values of Tc measured in heating and cooling was on the order of 10 mK, indicating that equilibrium between temperature sensors and sample was nearly achieved. The offset between the sensors and the GIC is probably the largest source of error in the determination of Tc for reasonably narrow superconducting transitions. This offset is also one of the most important sources of error in Hc2(T) measurements at low temperatures.
The magnetic field was supplied by a set of homemade water-cooled copper Helmholtz coils about 2 feet in diameter and separated by about the same distance. No difficulties with hysteresis were encountered in using this magnet. With the largest available power supply, a pair of Hewlett Packard Model 8012A supplies connected in series, the magnet could be swept up to a field of about 1000 oersteds at about 30 amperes, which was adequate for the C4KHg critical field measurements. The field was calibrated with a Hall probe against the output of a potentiometer on the sweep control of the power supply. This sweep control was built by Bruce Brandt of the National Magnet Laboratory. No effort was made to screen out the earth's field, as its contribution to the experimental errors was thought to be inconsequential overall. The largest source of inaccuracy in the critical field measurements was in the field calibration, as the exact value of the field inside the Helmholtz coils was position-sensitive, varying by as much as 10% over the distance of an inch. However, since the field versus potentiometer calibration is linear, any inaccuracies in sample position or calibration should change only the magnitude of the data curves, not their shape. Because of the possible errors in sample mounting and field calibration, it is estimated that the critical field at a given orientation and temperature could be off by as much as 15% in either direction. However, the ratio of the values of Hc2 at two different angles or temperatures is known considerably more accurately, perhaps to within about ± 5% of the measured value.
Most of the data described in this chapter were obtained using a 3He closed-cycle refrigeration system built by Mike Blaho of the National Magnet Laboratory. The inductance bridge was attached to a standard 3He probe loaded into a 3He refrigerator which was immersed into a 4He bath. The germanium resistor was glued directly to the bridge in order to insure good thermal contact to the sample. The recirculation system and refrigerator are of standard design.[20] Some of the data above 0.9 K were taken in a single-shot 4He evaporator cryostat. No discrepancy was found between these and the 3He experiments.
