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PES 415 / Phys 515 Solid State Physics LabSpring 2005

High Temperature Superconductivity

Superconductivity can no longer be regarded as a rare event. At least 36 elements and thousandsof compounds and alloys become superconducting under some conditions. While not unusual,superconducting behavior can not be explained by the common model which assumes that

electrons move through materials independently. The traditional model of superconductivity, theBCS theory, requires the electrons to be coupled into what are called Cooper pairs.

The properties of superconductors set them apart from ordinary behavior of materials. Theyexhibit no measurable DC resistivity. Currents which are started in superconductors continuewithout any decay. They have been observed to last for at least 2.5 years! These materials alsobehave like a perfect diamagnet. In weak applied magnetic fields, electrical surface currents areformed which precisely cancel the applied field inside the superconductor.

Superconducting behavior sets in below a critical temperature Tc. Traditional superconductorshave a very sharp transition from normal to superconducting behavior at this temperature. Hightemperature superconductors appear to have a somewhat more gradual transition over about 5 K.

The first superconductor was found in 1911 and through 1973 the transition temperatures thatwere found were all in the 4.2 - 23 K range. This required the use of liquid helium to attain thesetemperatures. In 1986 a rather different material, LaBaCuOx was found to be superconductingwith a transition temperature of 35 K. The following year, YBa2Cu3Ox was discovered to have atransition temperature of 90 K. This represented a tremendous breakthrough in superconductivityapplications because liquid nitrogen has a temperature of 77 K. These new materials could bemade superconducting by cooling them with relatively inexpensive liquid nitrogen rather thanliquid helium. Progress in developing new superconducting materials continues with criticaltemperatures over 120 K now reported.

Even for temperatures below Tc, superconductivity is lost if an applied magnetic field is toostrong. The critical field Hc at which superconducting behavior ceases is often in the range of 1 -

2000 Gauss. Superconductivity is also lost if the current passing through a superconductorexceeds a critical current Ic. The specific value of this critical current depends on the geometry ofthe superconducting material. Since currents produce magnetic fields, it is not surprising to findthat the critical current and critical magnetic field can be related.

With the development of high Tc superconductors, most students have now seen thedemonstration of a magnet levitated above a superconductor. The levitation comes about becauseof the Meissner effect. The magnetic field of the magnet cannot penetrate into the superconductor.An induced surface current is formed which opposes the applied field and repels the magnet. Themagnet will be levitated if the magnetic force is stronger then the gravitational force. Strong, lightweight rare earth magnets are used for these demonstrations.

These effects can be shown from electromagnetic theory. We begin by considering a perfectconductor (which may or may not be a superconductor). The electrons flowing through itexperience a force F = ma with no type of "friction" term. Writing this in terms of the propertiesof the electrons:

-qeE = m (dv/dt)

where qe and m are the charge and mass of the electron. E is the electric field in the conductor andv is the velocity of the electrons. We can rewrite this expression in terms of the current density J

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since J = -qevn where n is the number density of electrons.

dJ/dt = (nqe2/m) EThis can be combined with Faraday's law: x E = - B/t to yield

/t [ x J + (nqe2/m) B ] = 0

This tells us that the quantity in the brackets has no time dependence. This is a requirement forperfect conductors. London and London, however, discovered that superconductors obey a morerestrictive form of the equation. They found that the bracketed term not only has no timedependence but that the term has a time-independent value of 0. This leads directly to the Londonequation:

x J = - (nqe2/m) B

To develop a better physical understanding of the London equation we can use Ampere's Law x B = oJ to substitute in for J:

x ( x B) = - (onqe2/m) B

From the rules of vector calculus: x ( x B) = ( B) - 2B and from Maxwell'sequations, we know B = 0. The resulting equation is

2 B = (onqe2/m) B

Similarly we could derive an equation for the current density:

2J = (onqe2/m) J

Differential equations of this form have exponential solutions. In one dimension the solutionslook like

B(x) = B(0) e- cx

We can define L = 1/c which is called the London penetration depth. It is a measure of how farinto the superconductor B and J penetrate. Putting in values for the constants we find that L isabout 10-2 - 10-3 . As we indicated previously the currents and fields exist only at the surfaceof the material.

Measurement techniques

We are interested in measuring temperature and resistance in these experiments. Temperature willbe measured using a thermocouple. Thermocouples make use of the Seebeck effect. If twodifferent materials are connected at two places (like two long wires connected at the ends), acurrent will flow between the two junctions if the temperatures of the junctions are different. Thissuggests that a voltage difference exists between the junctions. We can break the loop andmeasure this potential difference. The voltage which we measure (typically mV) is related to thetemperature difference. If we know the reference temperature at one end (typically roomtemperature of a fixed reference temperature like the triple point of water) then we can measure thetemperature at the other end.

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material A

material B

I

I

V1 V2T1 T2

Figure 1. Thermocouple made from wires of two different materials.

Thermocouples from a variety of standard materials have been calibrated and temperature vs.voltage tables are available. They typically look something like this:

Deg K 0 1 2 3 4 5 6 7 8 9 Deg K

70 6.92 6.85 6.78 6.71 6.64 6.57 6.50 6.43 6.36 6.29 70 80 6.22 6.15 6.08 6.01 5.94 5.87 5.80 5.73 5.66 5.59 80

90 5.52 5.45 5.38 5.31 5.24 5.17 5.10 5.03 4.96 4.89 90100 4.82 4.75 4.68 4.61 4.54 4.47 4.40 4.33 4.26 4.19 100

To determine the temperature, find the measured voltage in the body of the table. Then look toeither end of the table to detemine what decade you are in and look up to the top of the table todetermine the degree within that decade. For instance, if you measured 5.94 mV you can find thatnumber in the table. The temperature associated with it is 84 K. Obviously you may need tointerpolate our round your voltage numbers to use the table.

The other parameter we need to measure is the resistance. We could just use two wires attached tothe sample with a resistance meter across them, but this gives us the total resistance of the wires,the contact points and the sample. A better design is the four point probe. This uses four wireswhich contact the sample in a line. We pass a current between the two end wires and measure thevoltage which develops between the two inner wires. From Ohms Law, we know R = V/I so wecan determine the resistance without any contribution from the wires.

V

current source

Experiments

The samples used in these experiments are YBa2Cu3Ox with the x 7. Since the elements

Y:Ba:Cu appear in the ratio 1:2:3, these materials are called "1-2-3" superconductors. Thesuperconductors are formed by measuring out compounds of these elements so that they will bepresent in this ratio. The materials are ground into a powder and formed under pressure into thedesired shape (a disk in our case). The disk is then baked in a well-ventilated kiln at about 1000Cfor 8-12 hours and then slowly cooled over 8-12 hours in the presence of air or pure oxygen. Ifthe sample does not get enough oxygen it will be green in color and will not superconduct. If ithas the proper amount of oxygen it will be black and will superconduct. The quality of thesuperconductor can be improved by grinding the disk again and repeating the baking and coolingsteps.

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Meissner Effect

Demonstrate the Meissner effect by cooling a superconducting disk with liquid nitrogen andplacing a magnet on it. HANDLE THE LIQUID NITROGEN WITH CARE !! It is extremelycold and can cause skin damage with fairly short exposures. DO NOT HANDLE THESUPERCONDUCTOR WITH YOUR HANDS ! Always use a non-magnetic (plastic, nylon, ...)tweezers when handling the superconductor or magnets. (The oil from your fingers could harmthe materials).

How high is the magnet levitated? Spin the magnet. Try pushing the magnet to one side.Observations are as important in experimental physics as measurements are!

Critical Temperature

The superconductors used in the remaining experiments are encased in metal to provide greaterthermal and mechanical stability. One sample has two leads coming out from a thermocouplewhich is positioned against the bottom of the superconducting disk. The manufacturer claims thatsome additional complications in the design make the thermocouple readings differ from standard

values. They have calibrated the thermocouple and provide the following table for converting mVreadings to degrees Kelvin. The thermocouple should read approximately -0.16 mV at roomtemperature (298 K) and +6.43 mV if the entire assembly is immersed in liquid nitrogen.

Conversion from mV to K for superconductor with only a thermocouple attached to it:

Deg K 0 1 2 3 4 5 6 7 8 9 Deg K

70 6.92 6.85 6.78 6.71 6.64 6.57 6.50 6.43 6.36 6.29 7080 6.22 6.15 6.08 6.01 5.94 5.87 5.80 5.73 5.66 5.59 8090 5.52 5.45 5.38 5.31 5.24 5.17 5.10 5.03 4.96 4.89 90

100 4.82 4.75 4.68 4.61 4.54 4.47 4.40 4.33 4.26 4.19 100

Procedure for an approximate determination of the critical temperature:

1. Check the calibration of the thermocouple by measuring the temperature at room temperatureand when the assembly is fully immersed in liquid nitrogen. This can be done while doing theexperiment.

2. Place the assembly (superconductor side up!) in a shallow insulating container with a magneton top of it. Add liquid nitrogen to the container.

3. Observe the magnet and record the temperature at which the magnet is levitated from thesurface.

4. Now allow the superconductor to warm up and record the temperature when the magnet returnsto the surface.

5. Are the two temperatures the same ? Why or why not ?

A more precise determination of the critical temperature can be made by using a superconductingsample with both a thermocouple and a 4-point probe for measuring resistance of the sample. Wecan now quantitatively observe when the resistance goes to zero and how it approaches zero. Aslightly different design has led to slightly different calibration factors for the thermocouple andthe revised conversion chart, provided by the manufacturer, is reproduced below:

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A schematic diagram of the device is provided below showing the connections for the four pointprobe and thermocouple.

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Procedure:

1. Connect a 0.5 amp DC power supply between the black wires. You may want to include a meterto get a more accurate measurement of the current. Measure the voltage from the 4-point probe atthe yellow wires. The temperature and resistance change rather quickly, so rather than take data byhand from voltmeters, you may want to use an x-y plotter with the temperature on the x-axis andvoltage on the y-axis. The voltages are small so use a mV scale. Be sure to mark the zero point soyou know what your scale is. You may also be able to interface these voltages to a computer.

2. Place the superconductor device in a shallow insulating container. Measure the current(constant), voltage, and temperature. As you cool the apparatus down with liquid nitrogen, recordthe temperature and voltage (by hand, x-y plotter, or computer). Once the temperature hasstabilized at liquid nitrogen temperature, remove the device from the liquid nitrogen and record thevoltage and temperature again as the assembly warms up. Since R = V/I you can now plot theresistance as a function of temperature.

Further Reading:

N. W. Ashcroft and N. D. Mermin, "Solid State Physics" (Holt, Rinehart and Winston, NewYork, 1976). - theory, Meissner Effect

D. D. Pollock, "Physics of Engineering Materials" (Prentice-Hall, Inc., Englewood Cliffs, NJ,1990). - theory and applications

A. Khurana, "Superconductivity seen above the boiling point of nitrogen", Physics Today 40, 17(April, 1987). - development of high temperature superconductors

E. A. Early, C. L. Seaman, K. N. Yang, and M. B. Maple, "Demonstrating superconductivity atliquid nitrogen temperatures", Am .J. Phys. 56, 617 (1988). - theory and experiment

A. M. Wolsky, R. F. Giese, and E. J. Daniels, "The New Superconductors: Prospects forApplications", Scientific American 260, 61 (February, 1989).

J. M. Rowell, "Superconductivity Research: A Different View", Phys. Today 41, 38 (Nov.,1988).R. A. Dunlap, Experimental Physics: Modern Methods thermocouples

many Modern Physics texts have discussions of superconductors.

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