# LTC Magnetization and Transition Temperatures of Low Temperature Superconductors

```University of Toronto
LTC
Magnetization and Transition
Temperatures of Low Temperature
Superconductors
Revisions:
2015 December:
1988 June:
David Bailey <[email protected] >
Derek Manchester and John Pitre
Please send any corrections, comments, or suggestions to the professor currently supervising this
experiment, the author of the most recent revision above, or the Advanced Physics Lab Coordinator.
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INTRODUCTION
A superconductor is a material that is both perfect conductor with zero resistance and a perfect
diamagnet1 expelling any magnetic field. Superconductivity is a quantum phenomena that can occur
in a normal metal if conduction electrons locally distort the atomic lattice in such a way that there is
a short range effective attractive force between electrons. If this attractive force is stronger than the
Coulomb repulsive force between a pair of electrons, weakly bound “Cooper pair” electron-electron
states can form. These pairs form when thermal energies (~kT) are less than the pair binding energy
2Eg, where Eg is the energy gap between an electron in a Cooper pair state and a normal conducting
ground state. Since Cooper pairs are bosons with integer spin, they form a Bose-Einstein condensate
quantum state that can’t be scattered by the defects and excitations that cause normal electrical
resistance, so the material has zero resistivity.
The energy gap Eg(T, H) depends on temperature (T) and magnetic field (H), and a material only
becomes superconducting below a critical temperature, TC. According to standard BCS theory, the
Cooper pair binding energy is 2Eg(T, H)≈7/2kTC.
Superconductors expel magnetic fields through the Meissner effect, but a sufficiently high magnetic
field can destroy the superconducting state. The magnetic field sufficient to force a bulk
superconductor to revert to its normal state at a temperature T is known as the critical field, Hc(T):
2
"T %
H c (T )
= 1− \$ '
(1)
H c (T = 0 )
# Tc &
Type I superconductors have a single critical field Hc, but Type II superconductors have two
apparent critical fields, Hc1 & Hc2, with (Hc1 Hc2)1/2 ≈ Hc. Below Hc1, all magnetic field is excluded as
for a Type I superconductor. Above Hc1 but below Hc2, magnetic field exists in “vortex” filamentary
normal regions within the superconducting material. The material is completely normal above Hc2.
Whether a superconductor is Type I or II depends on the ratio κ = λL/ξ. λL is the London penetration
depth which parameterizes how external magnetic fields are exponential damped below surface of a
superconductor. ξ0 is the intrinsic coherence length over which the superconducting electron density
cannot vary significantly, and is related to the typical separation of the electrons in a Cooper pair. A
superconductor is Type II when the penetration depth is greater than the coherence length. The
critical field values are related to κ:
Hc1 ≈ Φ0/(π λ2) ≈ Hc/κ,
Hc2 ≈ Φ0/(π ξ2) ≈ κHc
(2)
where Φ0=hc/(2e) is the minimum quantum of flux inside a superconductor.
For a further introduction and appreciation of the importance of magnetization measurements in
understanding and characterizing superconductors, see the Superconductivity Chapter in C. Kittel.
See C. Poole et al. or R.G. Sharma for further reading on Superconductivity.
EXPERIMENTAL
This experiment looks at the magnetization of superconductors as functions of the magnetic field and
of temperature. The critical fields (Hc, Hc1, Hc2) and transition temperature in zero field (Tc) are
measured and this provides enough information for all the important physical quantities related to
superconductivity in a metal to be determined.
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Note that perfect diamagnetism (interior field B=0) does not follow from zero resistivity, which
only implies dB/dt=0.
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The metals used in this experiment are lead (Pb), a Type I superconductor, and niobium (Nb), a Type
II superconductor. These metals both have transition temperatures just a few degrees above the
boiling point of liquid helium, 4.2 K. Using metals with Tc above rather than below 4.2 K avoids the
necessity of lowering the temperature of the superconductors by using cumbersome pumping
equipment.
Safety Reminders
• This experiment uses liquid helium, so all laboratory rules and precautions about handling this
liquid must be observed.
• Eye protection, gloves, and proper footwear, e.g. no sandals, must be worn when working with
cryogenic liquids or dewars.
• Sealed cryogenic containers build up pressure from the evaporating gas, so eye protection must
always be worn when opening the valves on the liquid helium dewar. You must ask for
instruction from the supervising professor, the Demo, or the Lab Technologist, when first using a
liquid helium dewar. Never leave the dewar with all valves closed for long periods; the safety
valve should prevent an explosion, but such a blow-out is not desirable.
• Filling the cryostat with liquid nitrogen and helium must be done under the supervision of the Lab
Technologist.
NOTE: This is not a complete list of every hazard you may encounter. We cannot warn against all
possible creative stupidities, e.g. juggling cryostats. Experimenters must use common sense to
assess and avoid risks, e.g. never open plugged-in electrical equipment, watch for sharp edges, don’t
lift too-heavy objects, …. If you are unsure whether something is safe, ask the supervising
professor, the lab technologist, or the lab coordinator. When in doubt, ask! If an accident or incident
happens, you must let us know. More safety information is available at
http://www.ehs.utoronto.ca/resources.htm.
Students should familiarize themselves with all the equipment before requesting liquid helium for
data acquisition. It normally takes about 6 hours to fill and cool the cryostat and take a series of
measurements. The cryostat must be reserved ahead of time by contacting the Lab Technologist.
The general layout of the coils for measuring the magnetization is shown in Figure 1. Copper wire
coils enclose 0.5 inch long plugs made from Lead (Pb, circular), Niobium (Nb, rounded square), and
Brass (triangle). The axis of the coils and plugs should be aligned with the external magnetic field.
Note that the coils are not infinitely thin and the Pb/Nb/Brass plugs only partially (by varying
amounts) fill their coils, so even when the Pb/Nb plugs are superconducting there still will be some
magnetic flux enclosed by the coils. The measuring coils are surrounded by a glass tube exchange
gas chamber connected through a 1/4 inch diameter tube to the probe top plate. A valve at the top
plate (Hoke) permits the connection of the glass chamber to the general internal space of the helium
dewar. When the helium dewar contains liquid helium, this valve may be used to connect the glass
chamber to the vapor over the liquid helium and thus opening this valve provides a simple way to
admit clean helium exchange gas to the glass chamber and thus a way of ensuring that the coils for
magnetization measurements are at the temperature of the liquid helium bath which is 4.2 K. For
magnetization measurements at temperatures above 4.2 K the Hoke valve on the top plate is closed
and the Veeco Valve which provides communication with a mechanical pump is opened. This valve
must not be opened when liquid helium is in the dewar unless the appropriately connected
mechanical pump is operating.
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The pressure/vacuum in the pumping line is indicated with sufficient precision by the dial gauge. A
reading of "30 vac" corresponds to a 30 inch height of a mercury column and is sufficient indication
that the mechanical pump is operating satisfactorily for the purpose of the experiment.
Temperature measurements are made using a silicon diode thermometer set in the copper block on
which the coils are mounted. The thermometer operates by passing a constant current from a
constant current source through the diode. The voltage required for this constant current is
temperature dependent. A calibration table and curves of voltage versus temperature for the diode
are given in Appendix I, along with the operating information for the constant current source.
Figure 1: Layout of the Coils on the Probe for Measuring Magnetization
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Connecting the glass chamber to the mechanical pump will provide sufficient vacuum insulation
around the copper mounting block containing the coils to allow the temperature of the measuring
block to rise. Probably some power in the heater will have to be used to aid this temperature change.
With a little care, the heater power can be adjusted to give a temperature which is stable enough to
make a measurement of the magnetization (time involved is about 2 minutes)
Terminal connections on the probe to the heater, diode and coils are given in Figure 2. The terminal
connections to the various coils are routed to a box and the terminal connections on the box are given
in Figure 3. Before any measurements are attempted, the circuitry for obtaining a response for the
magnetization should be set up. Use the connections to the measuring coil containing brass in an
arrangement which enables first the Nb coil and then the Pb coil to be connected in opposition to the
winding sense of the brass coil. This can be accomplished using female-female banana plug
couplings with the special leads provided. All of these coils have very closely the same number of
turns and therefore electrical resistances. Thus when they are connected in opposition, the induced
e.m.f. produced by "ramping" the magnetic field through the coils will be due to the effect of the
magnetization of Nb or Pb. NOTE: Because the system is sometimes repaired, it is possible that
some of the connections in Figure 3 could be reversed. By ramping the magnet, it can be checked
that the Brass coil is connected in opposition to the Nb and Pb coils, and if incorrect the leads can be
swapped.
Figure 2: Terminal Connections on the Probe
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Ramping the magnetic field is achieved by using the ramp generator connected to the modulation
input on the magnet power supply (it should be already connected). The output of the magnetization
coils can be fed through a milli-microvoltmeter to the Y terminals of the X-Y plotter. Try very low
gain settings to start with on both voltmeter and plotter in order to avoid overdriving the plotter. Do
as much of the circuit testing as possible at room temperature before transferring any liquid helium.
Figure 3: Terminal Connections to the Various Coils on the Box
INTERPRETATION OF THE RESULTS
By Faraday’s Law, the electromotive force (i.e. voltage) output of each coil is proportional to the rate
of change in the magnetic flux through the coil, so the magnetization results have to be integrated to
give an M(H) curve. The variation of M(H) with temperature can give a value of Tc together with
values of Hc(T), Hc1(T), Hc2(T) and possibly Hc1(0) and Hc2(0). Lead is a Type I superconductor and
thus only Hc(T), Hc(0) and Tc are obtained. With these experimental data available, determine all the
important parameters for the Type II superconductors that you can e.g. κ, ξ, λ, etc. For a Type I
superconductor, you do not have enough information to determine such parameters from
experimental values. Figure 5 of E.H. Brandt shows how the ideal magnetization, M(H) of long
Type II superconductors, depends on κ.
In interpreting your experimental data, note that you are looking at the magnetization of a short
cylinder. Figure 10 of E.H. Brandt shows how M(H) depends on the aspect ratio (diameter/length)
and exhibits hysteresis, i.e. differs for H increasing and decreasing.
REFERENCES
1. E.H. Brandt, The Vortex Lattice in Conventional and High-Tc Superconductors, Brazilian
Journal of Physics 32 (2002) 675-684; http://www.scielo.br/pdf/bjp/v32n3/a02v32n3.
2. C. Kittel, (1996) Introduction to Solid State Physics, 7th Edition, Wiley, 1996; QC171 .K5 1996.
3. C. Poole, H. Farach, R. Creswick and R. Prozorov, Superconductivity, 3rd Edition, Elsevier,
2014; http://www.sciencedirect.com.myaccess.library.utoronto.ca/science/book/9780124095090.
4. R.G. Sharma, Superconductivity: Basics and Applications to Magnets, Springer, 2015;
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APPENDIX I
CALIBRATION CURVES AND DATA FOR TEMPERATURE SENSITIVE SILICON
DIODE
Calibration Curve for Silicon Diode (S/N 17547) - 4.0 K → 300 K
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