Superconductivity: Measurement of the Critical Temperature

Truitt Wiensz and Adam Bourassa

February 2, 2001

1 Objectives

There are four main objectives:

a) A susceptibility probe will be used to measure the critical temperature of a

superconductor.

b) The Meissner effect will be used to measure the critical temperature of a

superconductor as a function of drive current.

c) The critical temperature will also be measured as a function of drive current

by direct measurement of the electrical resistance of the sample with a four

point probe.

d) The critical current will be determined for the superconductor at a

temperature of 77 K.

2 Theory

2.1 BCS Theory

In a material in the superconducting state, there are two types of electrons:

superconducting electrons and normal electrons. The density of the superconducting

electrons, sn , increases with temperature from the total electron density at KT 0= ,

to zero at the critical temperature, CTT = . A temperature-dependent energy gap,

∆ , which has a maximum value at KT 0= and goes to zero at CTT = , separates

the superconducting electrons from the normal electrons. Electromagnetic waves

2

incident on a superconductor are absorbed only when the photon energy is high

enough to excite the electrons above the energy gap, ∆ .

∆== 2νhEphoton (1)

Equation (1) contains the factor of 2 because superconducting electrons exist only in

a bound state called Cooper pairs; therefore the arriving photon must be energetic

enough to excite two electrons across the energy gap. It follows that equation (1)

describes the binding energy of the Copper pair.

In order to maximize the reduction in energy due to the attraction of the

superconducting electrons, all of the pairs must have the same wavefunctions. This

means that superconductivity is a cooperative phenomenon: all of the electrons are

Cooper pairs at KT 0= , and as temperature increases, some of the pairs are

broken from thermal excitation. This leads to a decrease in ∆ as CTT → .

For a uniform current density, the pairs must take on wavefunctions such as

rq ⋅Ψ=Ψ ie0 , (2)

where 0Ψ is a wavefunction for a pair at rest, r is the position of the center of mass

of the two electrons, and q corresponds to a centre-of-mass momentum of qh .

Thus for an electron mass, m , and a velocity of the pair, v ,

vq m2=h . (3)

Since a Cooper pair has a charge of e2− , the current density (the number of charge

carriers multiplied by the charge and the velocity) in the superconducting state is

m

ens

22

2q

jh

−= . (4)

3

This Cooper pair current is unaffected by phenomenon that would lead to a

resistance to the flow of the current such as the scattering of electrons by phonons

and by impurities. A Cooper pair will absorb a phonon of energy ∆2 and split to form

two electrons, but only at the same rate that two electrons will combine and emit a

phonon to form a Cooper pair with exactly the same wavefunction as those pairs that

already exist. Unless the center-of-mass motion of the secondary pair is the same

as that of the existing pairs, the binding energy of the pair must be zero (it is no

longer in a bound state). It is also for this reason that impurities cannot affect the

pair current. Since scattering from an impurity would mean a change in momentum

for one Cooper pair, the pair must lose its binding energy. Thus current of Cooper

pairs can only be affected by phenomenon like the electric field that changes all pairs

in the same way.

2.2 The Meissner Effect

Superconductors are separated into two types depending on how they are affected

by the presence of a magnetic field.

Type I superconductors lose their superconducting ability in the presence of a

magnetic field higher than the critical field, )(TBc . This means there will also be a

critical current, )(TIc , for which the magnetic field created by the current flow is

equal to )(TBc . When a material is in the superconducting state, screening currents

are induced on the surface of the conductor so as to generate a magnetic field that

exactly cancels the applied magnetic field. This expulsion of the magnetic flux is

called the Meissner Effect and can be explained by taking the superconductor to be a

magnetic material with a magnetization M . The expulsion of flux inside the

superconductor requires

0)(0 =+= MHB µ , (5)

4

so

HM −= . (6)

Since HM χ= , where χ is the magnetic susceptibility, a superconductor in the

Meissner state has a susceptibility of 1−=χ which means that it is a perfect

diamagnet.

Type II superconductors behave differently in the presence of an external magnetic

field, and have no relevance to this experiment.

2.3 The Critical Field BC

The superconductor used in this experiment, YBa2Cu3O7, is a Type I superconductor.

As such, its superconductivity is destroyed by a modest applied magnetic field. An

empirical result expressing the temperature dependence of the critical field is given

by the following:

−=

2

1)0()(C

CC TT

BTB , (7)

where TC is the critical temperature of the material.

3 Apparatus

3.1 Susceptibility Probe

The susceptibility probe consists of an approximately 1 cm diameter section of a

superconducting rod with 400 turns of wire around it. An AC current is introduced

into the coil causing a magnetic field to be generated. If the superconductor is cooled

below the critical temperature, it will expel the flux of the magnetic field created by

the coil, causing a distinct change in the coil's inductance. A thermocouple is also

placed on the probe to allow temperature measurement of the superconductor. The

circuit used in combination with the probe is shown in Figure 1.

5

Figure 1: Susceptibility probe circuit configuration

3.2 Four Point Probe

The four point probe consists of a superconducting sample with four leads connected

to it and a thermocouple placed on the superconductor. A schematic diagram of the

connection used is shown in Figure 2. Probes 1 and 4 are used to connect a dc

power supply so a constant current will flow through the length of the sample.

Probes 2 and 3 are connected to a voltmeter to measure the potential drop across

the length of the superconductor. Since the voltmeter has a high input impedance,

basically no current will flow through these leads so that the potential difference

does not include a drop due to the contact resistance of the leads (which would be a

factor if a simple ohmmeter was used). The resistance of the sample between these

leads is the ratio of the measured potential difference to the drive current.

6

Figure 2: Four point probe circuit configuration

4 Procedure

4.1 Measuring the Critical Temperature Using the Susceptibility Probe

After the circuit shown in Figure 1 was connected, the probe was placed in a cryostat

and covered with sand. Liquid nitrogen was added to the cryostat. The thermocouple

voltage, the ac current through the probe and the ac voltage across the probe were

periodically measured as the sample gradually warmed up. The same procedure was

repeated with an ohmmeter connected to leads of the probe so that the resistance of

the coil could be measured as the sample warmed. These measurements allow

calculation of the inductance of the coil (See equation 8, Section 6.1).

4.2 Measuring the Critical Temperature using the Meissner Effect

The superconducting sample was placed in the specially machined holder in the

cryostat and the cryostat was filled with liquid nitrogen. A dc power supply was

connected to the leads of the superconductor to supply a constant current. One of

the rare earth magnets was placed on the superconductor. As long as the sample

was cold enough to remain in the superconducting state, the magnet floated above

the superconductor. For various values of drive current, the sample was allowed to

7

gradually warm up. The temperature where the magnet no longer floated above the

superconductor was recorded using the thermocouple.

4.3 Measuring the Critical Temperature by Measurement of Resistance

The circuit shown in Figure 2 was connected and the superconductor was placed in

the cryostat filled with liquid nitrogen. Using the four point probe, a dc current

supply was connected. As the sample was allowed to gradually warm, the voltage

from the four point probe was recorded. The resistance is the ratio of the voltage to

the drive current.

4.4 Determining the Critical Current at 77 K

Using the four point probe to supply the measure of resistance by measuring the

current and voltage, the superconductor was placed in liquid nitrogen and kept

submerged so that the temperature remained at 77 K. The drive current was

increased until a non-negligible resistance was measured. The power supply that

was used could not supply current greater than 0.4 A. This was not enough current

at 77 K to destroy the superconducting state. An extrapolation of the results

obtained in Section 4.3 will be used to determine the critical current at this

temperature.

5 Observations

5.1 Critical Temperature Measurement - Susceptibility Probe

The thermocouple voltage, the ac current through the probe, the ac voltage across

the probe, and the probe resistance that were recorded as the sample gradually

warmed are tabulated in Appendix B.1. Rather than including plots of each of these

measurements, the impedance of the coil was determined with these measurements

8

directly by using equation 8 in Section 6.1. Figure 3 is plot of the calculated

impedance over the temperature range.

Temperature Dependence of Coil Impedance

8.00

9.00

10.00

11.00

12.00

13.00

14.00

75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0 120.0

Temperature (K)

Co

il Im

ped

ance

(O

hm

s)

Figure 3: Temperature dependence of coil impedance

5.2 Critical Temperature Measurement - Meissner Effect

The rare earth magnet was observed to float above the superconductor after it was

cooled using the liquid nitrogen. As the sample warmed the magnet ceased to float

at the critical temperature. This was repeated several times for various values of

drive current and the temperatures where the magnet “dropped” were recorded

(See Appendix B.2). Figure 4 is a graph of the observed “dropping” point

temperatures as a function of the drive current.

9

Perceived Magnet Dropping Temperature

84.0

86.0

88.0

90.0

92.0

94.0

96.0

98.0

100.0

102.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

Drive Current (A)

Cri

tica

l T

emp

erat

ure

(K

)

Figure 4: Perceived magnet dropping point as a function of drive current

5.3 Critical Temperature Measurement – Measuring Electrical Resistance

Using the four point probe, the resistance of the superconductor was measured for

various values of drive current. Figure 5 is a compilation graph of the measured

resistance as a function of temperature for the different values of drive current.

Error bars have been left off for clarity of presentation; tabulated values and

separate graphs of each current step, with error bars, have been included in

Appendices B and C respectively.

10

Transition Temperature - Master Chart

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0 120.0

Temperature (K)

4-p

oin

t pro

be

resi

stan

ce (m

illiO

hm

s)

I = 0.1 A

I = 0.2 A

I = 0.3 A

I = 0.4 A

Figure 5: Superconductor resistance over temperature for different drive currents.

6 Analysis

6.1 Critical Temperature Measurement - Susceptibility Probe

The conversion of the thermocouple voltage measurements to temperature in

degrees Kelvin was done using the conversion table supplied with the thermocouple.

Rather than doing linear interpolation between data points given in the table in order

to obtain temperature measurements, a least squares polynomial was fit to the

conversion table data, and then used for the conversion. A plot of the conversion

table and the polynomial fit is included in Appendix D.1. The polynomial fit the data

with a correlation of 1.0000.

Calibration of the thermocouple was tested by submerging it in liquid nitrogen (77

K), which yielded a thermocouple voltage of 6.38 mV, in contrast with the tabulated

11

value of 6.42 mV. The 0.04 mV error was carried through in error propagation, by

the calculus method, for determining temperature errors. All reported temperature

error bars were obtained in this fashion.

As described in the procedure, the resistance of the coil was measured as a function

of temperature. The same type of fitting procedure was performed on these

resistance values so that for any temperature where the voltage across and current

through the probe were measured, the resistance could be determined using the

fitted polynomial. The correlation of this fit was 0.9998. A plot of the resistance

values and the fit used is included in Appendix D.2. Due to the strong correlations

attained by the fitting functions, the calculus method was used in the standard way

on the fitting functions on the to obtain errors in the interpolated values.

Figure 3 is a plot of the calculated coil impedance as a function of temperature. The

inductance of the coil, L , was determined from this impedance, LZ , since LL ω=|| Z ,

where )1000(2 Hzπω = is the frequency of the ac signal. Using the inductance, the

magnetic susceptibility was calculated and plotted in Figure 6 using equation 11 in

this section.

12

Magnetic Susceptibility

-0.980

-0.960

-0.940

-0.920

-0.900

-0.880

-0.860

-0.840

-0.820

75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0 120.0

Temperature (K)

Pro

be

Su

scep

tib

ility

Figure 6: Magnetic susceptibility of the superconductor as a function of temperature

This same plot is included in Appendix C with fine grid lines added over the plot area.

It is evident from this plot that the transition from the superconducting region to the

normal region in the absence of drive current occurs between approximately 84 K

and 88 K. Error bars on Figure 6 were obtained by the calculus method from the

expression for the coil impedance,

( ) 22LLL RIVLZ +== ω , (8)

for the circuit shown in Figure 1. The expression for these errors then follows:

⋅+

⋅+

⋅= LL

LLL RRI

IVI

VVδ

δδδ 3

2

2L

L ZZ

1 (9)

13

Here VL and I represent measured coil voltage and drive current, and RL is the

temperature-dependent coil resistance. Coil inductance L may be obtained from the

definition of inductor impedance, ωLZ=L , giving the following error expression:

LZδω

δ1

=L . (10)

Magnetic susceptibility, χ, of the sample is given by:

−= 1

1

0LL

fχ , (11)

while its error, as displayed in Figure 6, is given by the following:

0

1LL

fδ

δχ = . (12)

Here L0 represents coil impedance as measured in vacuum, and f represents the

fraction of coil volume occupied by the superconducting sample. Equation 11 was

obtained under the assumption that f is close to unity, in that the sample occupies

the majority of the coil volume. Geometrical corrections to (11) are necessary in

order to better obtain the sample susceptibility.

6.2 Critical Temperature Measurement - Meissner Effect

Basically no quantitative analysis was required on the data collected by observing the

floating magnet. The critical temperature can be measured directly at the time that

the magnet was perceived to have dropped.

Error bars on the plot of this data (Figure 4) come directly from the precision of the

meters used to perform the measurements, and do not include any error due to the

ability to recognize the exact point where the magnet dropped. This was a very

difficult task to do consistently and accurately. As the superconductor warmed up,

certain areas on the surface of the superconductor (often near the edge) seemed to

14

be less effective at expelling the flux of the magnet and the magnet was moved to

the areas where the expulsion seemed to be the strongest. The superconductor was

obviously not at the same temperature throughout the material. Even in the areas

perceived to be the coldest, the magnet did not have a definitive dropping point.

The height that the magnet floated seemed to begin to decrease gradually until the

magnet was resting on a corner. After some time, it would slowly (or sometimes not

so slowly) lay flat on the superconductor. Even once this occurred, the magnet

seemed difficult to move with tweezers across the surface of the superconductor

indicating some type of magnetic field interaction. Basically the errors bars in

determining the critical temperature as the temperature where the magnet was

perceived to have dropped could be inordinately large.

6.3 Critical Temperature Measurement – Measuring Electrical Resistance

Included in Appendix C are four graphs that show the resistance measured using the

four-point probe for various values of drive current. Each of the curves is

approximately the same shape: the curve begins at very nearly zero resistance at

cold temperatures and at a certain temperature, begins to rise, indicating a

measurable resistance. We have considered the critical temperature to be the point

where the resistance first begins to rise from zero. On each of the four graphs in

Appendix C, the critical temperature is labeled and shown with an arrow. Figure 7 is

a plot of these critical temperatures as a function of drive current.

15

Critical Temperature versus Drive Current

76.0

78.0

80.0

82.0

84.0

86.0

88.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

Drive Current (A)

Cri

tica

l T

emp

erat

ure

(K

)

Figure 7: Critical temperature as a function of drive current (using direct measure of

resistance with four point probe)

Errors in drive current result from meter precision, and temperature errors were

obtained as discussed in Section 6.1. A second order polynomial fit was done on the

data shown in Figure 7, as suggested by equation 7 in the theory presented in

Section 2.3. The fit is shown in Figure 8.

16

Figure 8: Second order polynomial fit to critical temperature dependence on current

In Figure 8, the quadratic fit was extrapolated back in order to predict the value of

critical temperature for zero drive current. The value predicted by the fit is

approximately 87 K.

6.4 Determining the Critical Current at 77 K

As presented in Section 2.3, there exists an applied critical field BC, and thus a

critical coil current, below which the material is superconducting. The quadratic fit

shown in Figure 8 can also be used to determine the value of the critical current at

77 K, since the power supply that was used could only produce 0.40 A of current

which was not sufficient. The equation of the quadratic is:

T(I)= -37.5(I)2 - 5.95(I) + 87.175 (7)

For a temperature of 77 K, the root of this equation is 0.45 ± 0.03 A.

Critical Temperature versus Drive Current

74.0

76.0

78.0

80.0

82.0

84.0

86.0

88.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Drive Current (A)

Cri

tical

Tem

pera

ture

(K)

17

7 Discussion

7.1 Conclusions

Several methods of measuring the critical temperature of the sample were examined

in this experiment. Measuring the critical temperature of the sample, which carried

no current, with the susceptibility probe yielded a transition range from 84 K to 88 K.

This also provided a measurement of the susceptibility of the material in the

superconducting state, roughly –0.960 ± 0.002, in comparison with the ideal value

of –1. Some discrepancy in this result may be explained by the geometric

assumptions in equation 11, Section 6.1.

Examination of the critical field temperature dependence was performed with the

Meissner effect. The general trend of a lower critical temperature for a higher value

of drive current was observed, as can be seen in Figure 4. Results obtained in this

section were not consistent with expected results in several ways. First, the

temperature measurements of critical current were consistently offset, due to the

large error bars associated with subjective measurement, as discussed in Section

6.2. Also, the general trend of the critical current plot disagreed with expected

results (equation 7, Section 2.3), in that the concavity is reversed. Generally, flux

expulsion by the superconductor was observed for cold temperatures.

The four-point probe provided a measure of the dependence of critical temperature

on drive current. Results of this can be seen in Figures 7 and 8. The functional

dependence, including the negative concavity, as predicted in Section 2.3, was

observed in the temperature dependence of critical current. The predicted quadratic

relationship was fitted to the data, resulting in an extrapolated critical current of 0.45

± 0.03 A at a temperature of 77 K. Also, the fitting function provided an

extrapolated measure of the critical temperature at zero drive current. This was

18

found to be roughly 87 K, in agreement with the result of the susceptibility probe

measurement.

7.2 Points of Interest

Figure 5 is a plot of the superconductor resistance over temperature for different

values of drive current. In general, for each value of drive current, the

superconductor begins with essentially zero resistance and begins to increase in

resistance at a certain temperature. During the transition from the superconducting

state, the resistance depends proportionally on the value of drive current. As the

temperature approached a value around 103 K, the curves converge to a common

resistance as expected, since resistance in the normal state should not depend on

current. However, the transition to the normal state seemed occur over a

temperature range of over 20 degrees K, which is much larger than was observed

using the susceptibility probe. The transition was also wider for larger values of

drive current. We suspect that driving the superconductor with a significantly large

amount of current causes uneven resistive heating of the material and a

systematically longer transition from superconducting to normal state. This would

also explain some of the extremely hot critical temperatures and the uneven flux

expulsion observed using the floating magnet.

7.2 Suggestions for Improvement

We would like to suggest that this experiment be performed with the four point

probe using several values of current less than 0.1 A to determine if in fact resistive

heating of the superconductor during the transition to the normal state is extended

or oddly affected in some way.

19

The floating magnet experiment could be better performed if the experimenters

could identify the very first change in the height of the floating magnet rather than

trying to identify the point at the very end of the floating stage as we have done.

A better schematic of the superconductor used with the four point probe would have

been helpful in order to see how the drive current was applied to the geometry of the

material.

The fast warming of the superconductor did not allow for ease of gathering data,

especially during transition times. A computer to sample the values at small time

intervals or a more insulated system (perhaps even placing the system in a small

freezer) would help greatly with this problem.

20

Appendix

A. Equipment Listing

Quantity Equipment

1 Xantrex HPD 15-20 Power supply

1 WaveTek 27XT Digital Multimeter

2 Hewlett Packard 3468A Digital Multimeters

1 Susceptibility probe

1 Four-point probe

1 Hewlett Packard 3311A Signal Generator

21

B. Data Tables

B.1 Susceptibility Probe Measurement of Critical Temperature

Fraction of coil volume occupied by sample:f = 0.858

Coil inductance in vacuum:L_0 = 7.959 mH

Drive Coil Thermocouple Temperature Coil Coil Coil Probe Current Voltage Voltage Resistance Impedance Inductance Susceptibility(mA) (mV) (mV) (K) (Ohms) (Ohms) (mH)

Error: 0.01 0.005 0.004 1.0 0.101.49 5.830 6.387 75.6 7.79 8.72 +/- 0.10 1.388 +/- 0.016 -0.962 +/- 0.0021.48 5.820 6.378 75.8 7.79 8.73 +/- 0.10 1.389 +/- 0.016 -0.962 +/- 0.0021.48 5.830 6.326 77.2 7.78 8.72 +/- 0.10 1.388 +/- 0.016 -0.962 +/- 0.0021.48 5.840 6.317 77.4 7.78 8.72 +/- 0.10 1.388 +/- 0.016 -0.962 +/- 0.0021.48 5.850 6.259 78.9 7.78 8.72 +/- 0.10 1.388 +/- 0.016 -0.962 +/- 0.0021.48 5.860 6.230 79.7 7.78 8.73 +/- 0.10 1.389 +/- 0.016 -0.962 +/- 0.0021.48 5.870 6.217 80.0 7.78 8.73 +/- 0.10 1.389 +/- 0.016 -0.962 +/- 0.0021.48 5.880 6.198 80.5 7.78 8.74 +/- 0.10 1.390 +/- 0.016 -0.962 +/- 0.0021.48 5.890 6.188 80.8 7.78 8.74 +/- 0.10 1.391 +/- 0.016 -0.961 +/- 0.0021.48 5.900 6.176 81.1 7.78 8.75 +/- 0.10 1.392 +/- 0.016 -0.961 +/- 0.0021.48 5.920 6.162 81.5 7.79 8.75 +/- 0.10 1.393 +/- 0.016 -0.961 +/- 0.0021.48 5.940 6.146 81.9 7.79 8.76 +/- 0.10 1.395 +/- 0.016 -0.961 +/- 0.0021.48 5.960 6.139 82.1 7.79 8.77 +/- 0.10 1.396 +/- 0.016 -0.961 +/- 0.0021.48 5.980 6.133 82.3 7.79 8.78 +/- 0.10 1.397 +/- 0.016 -0.960 +/- 0.0021.48 6.000 6.126 82.5 7.80 8.79 +/- 0.10 1.399 +/- 0.016 -0.960 +/- 0.0021.48 6.020 6.120 82.6 7.80 8.80 +/- 0.10 1.400 +/- 0.016 -0.960 +/- 0.0021.48 6.100 6.101 83.1 7.81 8.83 +/- 0.10 1.405 +/- 0.016 -0.959 +/- 0.0021.48 6.200 6.095 83.3 7.81 8.86 +/- 0.10 1.410 +/- 0.016 -0.959 +/- 0.0021.48 6.300 6.085 83.6 7.81 8.90 +/- 0.10 1.416 +/- 0.016 -0.958 +/- 0.0021.48 6.400 6.082 83.7 7.81 8.93 +/- 0.10 1.421 +/- 0.016 -0.957 +/- 0.0021.48 6.700 6.077 83.8 7.81 9.03 +/- 0.10 1.437 +/- 0.016 -0.955 +/- 0.0021.47 7.000 6.066 84.1 7.82 9.16 +/- 0.10 1.457 +/- 0.017 -0.952 +/- 0.0021.47 7.500 6.049 84.5 7.83 9.34 +/- 0.10 1.487 +/- 0.017 -0.947 +/- 0.0021.47 8.000 6.036 84.9 7.83 9.54 +/- 0.11 1.518 +/- 0.017 -0.943 +/- 0.0021.47 9.000 6.020 85.3 7.84 9.95 +/- 0.11 1.583 +/- 0.017 -0.933 +/- 0.0021.47 10.000 6.009 85.6 7.85 10.39 +/- 0.11 1.653 +/- 0.017 -0.923 +/- 0.0031.47 12.000 6.000 85.9 7.85 11.33 +/- 0.11 1.803 +/- 0.018 -0.901 +/- 0.0031.47 13.000 5.988 86.2 7.86 11.83 +/- 0.11 1.883 +/- 0.018 -0.889 +/- 0.0031.47 14.000 5.968 86.8 7.87 12.36 +/- 0.12 1.967 +/- 0.019 -0.877 +/- 0.0031.47 14.700 5.890 88.9 7.93 12.76 +/- 0.12 2.031 +/- 0.019 -0.868 +/- 0.0031.47 14.730 5.828 90.6 7.98 12.81 +/- 0.12 2.039 +/- 0.019 -0.867 +/- 0.0031.47 14.760 5.782 91.9 8.02 12.85 +/- 0.12 2.045 +/- 0.019 -0.866 +/- 0.0031.47 14.800 5.714 93.8 8.08 12.91 +/- 0.12 2.055 +/- 0.019 -0.864 +/- 0.0031.47 14.825 5.677 94.8 8.12 12.95 +/- 0.12 2.061 +/- 0.019 -0.863 +/- 0.0031.47 14.900 5.554 98.3 8.24 13.07 +/- 0.12 2.079 +/- 0.019 -0.861 +/- 0.0031.47 14.975 5.445 101.3 8.36 13.18 +/- 0.12 2.097 +/- 0.019 -0.858 +/- 0.0031.47 15.000 5.417 102.1 8.38 13.21 +/- 0.12 2.102 +/- 0.019 -0.857 +/- 0.003

22

B.2 Meissner Effect Measurement of Critical Temperature

Drive Thermocouple TemperatureCurrent Voltage(A) (mV) (K)

Error: 0.01 0.005 1.00.40 5.960 87.00.35 5.950 87.20.30 5.910 88.30.25 5.875 89.30.20 5.775 92.10.15 5.750 92.80.10 5.630 96.10.05 5.540 98.7

23

B.3 Electrical Resistance Measurement of Critical Temperature

Trial (1) Trial (2)Drive Current: Drive Current:

0.10 +/- 0.01 A 0.20 +/- 0.01 A4-pt Voltage Probe Thermocouple Temperature 4-pt Voltage Probe Thermocouple Temperature

Resistance Voltage Resistance Voltage(mV) (mOhms) (mV) (K) (mV) (mOhms) (mV) (K)

Error: 0.002 1.0 Error: 0.002 1.00.003 0.03 +/- 0.02 6.335 76.9 0.008 0.04 +/- 0.01 6.350 76.60.004 0.04 +/- 0.02 6.303 77.8 0.007 0.04 +/- 0.01 6.300 77.90.003 0.03 +/- 0.02 6.250 79.2 0.007 0.04 +/- 0.01 6.275 78.50.003 0.03 +/- 0.02 6.200 80.5 0.008 0.04 +/- 0.01 6.250 79.20.003 0.03 +/- 0.02 6.150 81.8 0.009 0.05 +/- 0.01 6.225 79.80.003 0.03 +/- 0.02 6.100 83.2 0.015 0.08 +/- 0.01 6.175 81.20.002 0.02 +/- 0.02 6.030 85.1 0.025 0.13 +/- 0.02 6.125 82.50.010 0.10 +/- 0.03 5.990 86.2 0.050 0.25 +/- 0.02 6.050 84.50.018 0.18 +/- 0.04 5.940 87.5 0.075 0.38 +/- 0.03 6.012 85.60.025 0.25 +/- 0.05 5.920 88.1 0.100 0.50 +/- 0.04 5.978 86.50.050 0.50 +/- 0.07 5.870 89.4 0.125 0.63 +/- 0.04 5.958 87.00.075 0.75 +/- 0.10 5.830 90.6 0.150 0.75 +/- 0.05 5.941 87.50.100 1.00 +/- 0.12 5.780 91.9 0.200 1.00 +/- 0.06 5.908 88.40.125 1.25 +/- 0.15 5.750 92.8 0.250 1.25 +/- 0.07 5.880 89.20.150 1.50 +/- 0.17 5.730 93.3 0.300 1.50 +/- 0.09 5.847 90.10.175 1.75 +/- 0.20 5.720 93.6 0.350 1.75 +/- 0.10 5.820 90.80.200 2.00 +/- 0.22 5.690 94.5 0.400 2.00 +/- 0.11 5.784 91.80.250 2.50 +/- 0.27 5.660 95.3 0.450 2.25 +/- 0.12 5.754 92.70.275 2.75 +/- 0.30 5.625 96.3 0.500 2.50 +/- 0.14 5.724 93.50.300 3.00 +/- 0.32 5.600 97.0 0.600 3.00 +/- 0.16 5.672 95.00.350 3.50 +/- 0.37 5.560 98.1 0.700 3.50 +/- 0.19 5.623 96.30.400 4.00 +/- 0.42 5.510 99.5 0.800 4.00 +/- 0.21 5.567 97.90.450 4.50 +/- 0.47 5.470 100.6 0.900 4.50 +/- 0.24 5.510 99.50.500 5.00 +/- 0.52 5.440 101.5 1.000 5.00 +/- 0.26 5.450 101.20.550 5.50 +/- 0.57 5.400 102.6 1.100 5.50 +/- 0.29 5.393 102.80.600 6.00 +/- 0.62 5.350 104.0 1.200 6.00 +/- 0.31 5.330 104.60.645 6.45 +/- 0.67 5.190 108.6 1.250 6.25 +/- 0.32 5.240 107.20.650 6.50 +/- 0.67 5.150 109.7 1.265 6.33 +/- 0.33 5.140 110.0

24

Trial (3) Trial (4)Drive Current: Drive Current:

0.30 +/- 0.01 A 0.40 +/- 0.01 A4-pt Voltage Probe Thermocouple Temperature 4-pt Voltage Probe Thermocouple Temperature

Resistance Voltage Resistance Voltage(mV) (mOhms) (mV) (K) (mV) (mOhms) (mV) (K)

Error: 0.002 1.0 Error: 0.002 1.00.030 0.10 +/- 0.01 6.270 78.6 0.060 0.15 +/- 0.01 6.430 74.50.032 0.11 +/- 0.01 6.242 79.4 0.060 0.15 +/- 0.01 6.320 77.30.040 0.13 +/- 0.01 6.220 80.0 0.075 0.19 +/- 0.01 6.265 78.80.050 0.17 +/- 0.01 6.197 80.6 0.100 0.25 +/- 0.01 6.217 80.00.075 0.25 +/- 0.02 6.143 82.0 0.150 0.38 +/- 0.01 6.150 81.80.100 0.33 +/- 0.02 6.088 83.5 0.200 0.50 +/- 0.02 6.097 83.20.125 0.42 +/- 0.02 6.055 84.4 0.300 0.75 +/- 0.02 6.044 84.70.150 0.50 +/- 0.02 6.027 85.1 0.400 1.00 +/- 0.03 6.000 85.90.200 0.67 +/- 0.03 6.000 85.9 0.500 1.25 +/- 0.04 5.975 86.60.250 0.83 +/- 0.03 5.967 86.8 0.600 1.50 +/- 0.04 5.945 87.40.300 1.00 +/- 0.04 5.950 87.2 0.700 1.75 +/- 0.05 5.916 88.20.400 1.33 +/- 0.05 5.910 88.3 0.800 2.00 +/- 0.06 5.883 89.10.500 1.67 +/- 0.06 5.874 89.3 0.900 2.25 +/- 0.06 5.850 90.00.600 2.00 +/- 0.07 5.837 90.4 1.000 2.50 +/- 0.07 5.825 90.70.700 2.33 +/- 0.08 5.800 91.4 1.200 3.00 +/- 0.08 5.770 92.20.800 2.67 +/- 0.10 5.763 92.4 1.400 3.50 +/- 0.09 5.710 93.90.900 3.00 +/- 0.11 5.724 93.5 1.600 4.00 +/- 0.11 5.645 95.71.000 3.33 +/- 0.12 5.684 94.6 1.800 4.50 +/- 0.12 5.555 98.21.100 3.67 +/- 0.13 5.636 96.0 2.000 5.00 +/- 0.13 5.434 101.71.200 4.00 +/- 0.14 5.592 97.2 2.200 5.50 +/- 0.14 5.352 104.01.400 4.67 +/- 0.16 5.490 100.1 2.400 6.00 +/- 0.16 5.250 106.91.600 5.33 +/- 0.18 5.386 103.0 2.500 6.25 +/- 0.16 5.092 111.41.800 6.00 +/- 0.21 5.250 106.9 2.525 6.31 +/- 0.16 5.003 113.91.840 6.13 +/- 0.21 5.130 110.3 2.550 6.38 +/- 0.16 4.890 117.11.875 6.25 +/- 0.22 4.980 114.6

B.4 Determining the Critical Current at 77 K

Transition Drive Temperature Current(K) (A)

Error: 1.0 0.0286.2 0.1084.5 0.2082.0 0.3078.8 0.40

25

C. Plotted Data

Temperature Dependence of Coil Impedance

8.00

9.00

10.00

11.00

12.00

13.00

14.00

75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0 120.0

Temperature (K)

Co

il Im

ped

ance

(O

hm

s)

Magnetic Susceptibility

-0.980

-0.960

-0.940

-0.920

-0.900

-0.880

-0.860

-0.840

-0.820

75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0 120.0

Temperature (K)

Pro

be

Su

scep

tib

ility

26

Probe Resistance: I = 0.1 A

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

70.0 75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0

Temperature (K)

Res

ista

nce

(m

illiO

hm

s)

Critical Temp

Probe Resistance: I = 0.2 A

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

70.0 75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0

Temperature (K)

Res

ista

nce

(m

illiO

hm

s)

Critical Temp

27

Probe Resistance: I = 0.3 A

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

70.0 75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0 120.0

Temperature (K)

Res

ista

nce

(m

illiO

hm

s)

Critical Temp

Probe Resistance: I = 0.4 A

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

70.0 75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0 120.0

Temperature (K)

Res

ista

nce

(m

illiO

hm

s)

Critical Temp

28

D. Calibration Fitting Functions

D.1 Thermocouple Calibration

Following is a thermocouple calibration curve of temperature as a function of

thermocouple voltage, based on the data tables given in the supplementary

materials.

Thermocouple Calibration Polynomial

Fitted Temperature - Voltage Function (V in mV) :T(V) = 0.0043V6 - 0.0859V5 + 0.6743V4 - 2.8066V3 + 8.3245V2 - 50.654V + 289.79

Correlation Value:R2 = 1

0

50

100

150

200

250

300

350

-0.20 0.80 1.80 2.80 3.80 4.80 5.80

Thermocouple Voltage (mV)

Tem

per

atu

re (K

)

Interpolation was required in the temperature dependence of coil resistance for

analysis in Section 6.1. Following are the measured values of temperature-

dependent coil resistance, along with the fitted sixth-order polynomial describing this

fit.

29

D.2 Coil Resistance Temperature Calibration - Polynomial Fitting

Coil Thermocouple TemperatureResistance Voltage(Ohms) (mV) (K) 6.350 5.314 105.1

0.050 0.005 1.0 6.400 5.285 105.94.600 6.385 75.7 6.450 5.256 106.74.650 6.244 79.3 6.500 5.230 107.44.700 6.261 78.9 6.550 5.202 108.24.750 6.264 78.8 6.600 5.178 108.94.800 6.255 79.0 6.650 5.153 109.64.885 6.147 81.9 6.700 5.118 110.64.900 6.137 82.2 6.750 5.089 111.54.950 6.085 83.6 6.800 5.066 112.15.000 6.032 85.0 6.850 5.038 112.95.050 5.993 86.1 6.900 5.005 113.95.100 5.959 87.0 6.950 4.979 114.65.150 5.941 87.5 7.000 4.950 115.45.200 5.907 88.4 7.050 4.921 116.35.250 5.874 89.3 7.100 4.891 117.15.300 5.858 89.8 7.150 4.863 117.95.350 5.830 90.6 7.200 4.836 118.75.400 5.805 91.2 7.250 4.803 119.65.450 5.780 91.9 7.300 4.774 120.55.500 5.757 92.6 7.350 4.744 121.35.550 5.728 93.4 7.400 4.715 122.25.600 5.704 94.1 7.450 4.685 123.05.650 5.678 94.8 7.500 4.661 123.75.700 5.652 95.5 7.550 4.623 124.85.750 5.630 96.1 7.600 4.593 125.75.800 5.601 96.9 7.650 4.562 126.65.850 5.574 97.7 7.700 4.531 127.55.900 5.553 98.3 7.750 4.492 128.65.950 5.521 99.2 7.800 4.464 129.46.000 5.501 99.8 7.850 4.436 130.26.050 5.475 100.5 7.900 4.400 131.36.100 5.448 101.3 7.950 4.372 132.16.150 5.420 102.1 8.000 4.344 132.96.200 5.394 102.8 8.050 4.310 133.96.250 5.367 103.6 8.100 4.291 134.46.300 5.339 104.3 8.150 4.242 135.9

30

Coil Resistance Fitting Polynomial

Coil Resistance Fitting Function (V in mV) :

R(V) = -0.1461V6 + 4.7017V5 - 62.609V4 + 441.74V3 - 1742.3V2 + 3642.7V - 3144.1Correlation Value:

R2 = 0.9997

4.500

4.700

4.900

5.100

5.300

5.500

5.700

5.900

6.100

6.300

6.500

5.300 5.500 5.700 5.900 6.100 6.300

Thermocouple Voltage (mV)

Co

il R

esis

tan

ce (

Oh

ms)