|Department of Physics, Middlebury College||1992-93|
|Modern Physics Laboratory|
The term superconductivity refers to a number of unusual electric and magnetic phenomena that occur in many materials at low temperature.1,2,3 Although most experimental observables associated with the superconducting state can be described in the phenomenological language of classical electromagnetism, superconductivity is fundamentally a quantum mechanical phenomenon. The highly successful Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity developed in 1957 gives a fully quantum mechanical account of the existence and stability of the superconducting state in matter.
In this experiment you will investigate only two of the many important properties of the superconducting state.
(1) Complete absence of electrical resistivity.
In 1911, only three years after building the first helium liquifier, H. Kamerlingh-Onnes discovered that mercury loses its electrical resistance entirely when cooled below 4.2 K in a liquid helium bath. By vacuum pumping the helium gas above the liquid helium surface, he was able to attain temperatures near 1 K and found that a complete loss of electrical resistivity was possible for many pure elemental metals in the 1 to 9 K temperature range. Fig. 1 illustrates this loss of resistance for pure and impure specimens of tin (Sn) metal. The temperature at the midpoint of the transition from the normal state to the superconducting state is called the critical or transition temperature Tc. From Fig. 1, one can see that the effect of dilute impurities is to smooth the behavior of the resistivity in the transition region and to slightly increase the value of Tc.
Superconductivity can be destroyed by applying a sufficiently high external magnetic field. At fixed temperature T, a material will make a transition from the superconducting state to the normal state if an external magnetic field exceeding a critical field Hc(T) is applied. The variation of the critical field with temperature is shown in Fig. 2 for several elemental metals. The dependence of the critical field Hc on temperature is found empirically to obey
Hc(T) = Ho [ 1 - (T/Tc)2 ] (1)
where HO is the critical field at T = 0 K. Table I gives values of Tc and HO for several pure metals.
|TABLE I. Superconducting transition temperature Tc and critical field HO for some superconducting metals, from Ref. 4.|
(2) Complete exclusion of magnetic flux (Meissner effect).
When a magnetic field Ba is applied to a sample in the superconducting state, it is observed experimentally that the sample expels the applied field completely, so that B = 0 throughout the interior of the sample. This property of the superconducting state is known as the Meissner effect. Figs. 3(a)-(d) and 3(e)-(g) show that magnetic flux exclusion in the superconducting state takes place regardless of the order in which one performs the operations of cooling a sample and applying an external magnetic field.5 The applied magnetic field is cancelled everywhere in the interior of the superconductor by the opposing fields of what is called a screening supercurrent that exists in the superconducting sample.
It is tempting to rephrase property (1) by saying that a superconductor is a "perfect conductor". Then one could claim that property (1) implies property (2) because perfectly compensating induced currents would exactly oppose any attempt to put flux through the interior of a "perfect conductor". Unfortunately, this line of reasoning is false. Although a superconductor exhibits a complete absence of electrical resistance, it does not exhibit all of the properties necessary for it to be considered the limiting case of a "perfect conductor" in classical electromagnetism. To see this, consider the processes shown in Fig. 4 for a hypothetical "perfect conductor". The behavior of the "perfect conductor" in Figs. 4(a)-(d) is identical to what is observed for the behavior of a superconductor in Figs. 3(a)-(d); however, the sequence of steps in Figs. 4(e)-(g) demonstrate that a "perfect conductor" does not behave as a superconductor does. In Fig. 4(e) a sample of the hypothetical "perfect conductor" is placed in an applied magnetic field Ba at room temperature. In the normal state, the applied field is able to penetrate the interior of the sample. In Fig. 4(e)-(f) the sample is cooled below its transition temperature Tc. This cooling does not induce currents in the "perfect conductor" and the applied field Ba remains in the interior of the sample, as shown in Fig. 4(f). If the applied field Ba is then turned down to zero, as shown in Fig. 4(g), induced currents (Faraday's Law) flow without resistance to exactly maintain (Lenz's Law) the applied field Ba that had previously penetrated the interior of the "perfect conductor". Comparing Figs. 4(f) and 4(g) to Figs. 3(f) and 3(g) one finds that the experimentally observed properties of a superconductor are not compatible with those of a "perfect conductor".
If the above argument is too subtle to follow in a first reading, take comfort in the fact that superconductivity had been investigated for more than 20 years before Meissner and Oschenfeld made the first observations of magnetic flux exclusion in a superconductor in 1933 and clearly distinguished the experimentally observed behavior of superconductors from the behavior expected of a "perfect conductor". Since that time even more unusual low temperature phenomena have been observed, including magnetic flux quantization in a superconducting ring, quantum mechanical interference effects (Josephson junction), and superfluidity in quantum liquids.3,4
In the laboratory you will demonstrate the two superconductivity properties discussed above.
(1) Loss of electrical resistivity in Sn metal near 3.7 K.
Temperatures below 4.2 K can be achieved by immersing a sample in liquid helium and vacuum pumping the vapor above the liquid helium surface. In principle, if the helium vapor pressure could be varied over the range of pressure from 760 mm Hg (atmospheric pressure), down to 0.1 mm Hg, a temperature range from 4.2 K down to about 1 K would be accessible. In practice, it takes a fairly large mechanical pump to achieve pressures much below 100 mm Hg in the dewar of the present experiment. To "scan" the temperature near the superconducting transition for Sn metal, you will first lower the pressure in a liquid helium dewar to achieve a temperature near 3.0 K and then permit the dewar to warm up slowly through the Sn transition temperature region. A single temperature scan will take nearly four hours, so plan your work schedule accordingly.
Fig. 5 shows the various electrical connections on the sample probe and how they are routed to the instruments to be used to measure the resistance of the Sn wire and the temperature of the liquid helium bath. Before pumping the dewar to a temperature near 3.0 K, be sure that the Sn wire circuit is not open or short-circuited. Also, check that the calibrated Ge resistance thermometer gives a temperature near 4.2 K when the helium dewar is at atmospheric pressure. Use the Hall effect gaussmeter to check that the Helmholtz coil system is functioning properly.
When all of the instrumentation appears to be functioning properly, proceed with the following steps.
(a) Turn on the mechanical pump and open the valve to the liquid helium dewar. Pump the dewar for 30 minutes or until a temperature below 3.0 K has been reached according to the Ge resistance thermometer. During this process you should observe that the resistance of the Sn sample falls dramatically, indicating a transition from the normal state to the superconducting state.
When the pumping is complete, close the valve between the dewar and the mechanical pump and turn off the mechanical pump.
(b) Adjust the lock-in amplifier sensitivity to a more sensitive scale, so that noise gives at least a 0.1 V root-mean-square (rms) indication on the lock-in amplifier output meter. Start the strip chart and immediately record the time from a reliable clock.
- (c) At 15 minute intervals, as the dewar warms up, record:
- (i) time,
- (ii) Ge thermometer voltage and corresponding temperature,
- (iii) dewar pressure in mm Hg,
- (iv) rms signal amplitude from the lock-in amplifier (be sure to note if the sensitivity range switch is changed),
- (v) Helmholtz coil current necessary to drive the Sn sample to the normal state.
(d) When the dewar temperature reaches 4.0 K you may stop recording data. Be sure to record all settings of the various instruments before leaving the laboratory.
(2) Meissner effect in YBa2Cu3O7 at 77 K .
An elegant demonstration of the superconducting state may be seen in Fig. 6 where a permanent magnet is shown floating above a sample of YBa2Cu3O7 which has been cooled to 77 K, the normal boiling point of liquid nitrogen. The levitation of the magnet is due to the Meissner effect discussed above. The transition temperature Tc of this sample is estimated to be about 93 K, a temperature more than four times greater than had been measured for any material prior to April, 1987. This sample is known to have zero resistivity, yet it was made by heating a mixture of insulating and semiconducting metal oxides. Indeed, this is a very exotic material.
The exact composition of the sample shown in Fig. 6 is not known. The chemical formula YBa2Cu3Oz , where the subscript z indicates the number of oxygen atoms, is often used because the retention of oxygen by these samples depends strongly on the heat treatment used to make them. At elevated temperatures, atoms of oxygen exchange freely between the heated sample and the atmosphere and therefore the value of z depends on the means used to control the oxygen exchange. The sample shown in Fig. 6 is thought to have a value of z near 7 because of the marked similarity between its powder x-ray diffraction pattern6 and a published pattern identified as belonging to a double-layered structure of Y2Ba4Cu6O14+y.7 It is difficult to determine the exact structure of the superconducting material because there is disorder in the location of the oxygen atom vacancies and the x-ray diffraction pattern results from an average over the distribution of those vacancies. The superconducting properties of YBa2Cu3Oz samples are found to depend strongly on both the amount of oxygen and its distribution within a sample.
When a substance becomes superconducting, it dynamically excludes magnetic fields from the bulk of the superconducting region; this is the Meissner effect discussed earlier. The superconducting currents that act to screen the interior from magnetic fields are typically located within a few nm of the surface, so that virtually the entire volume of the superconductor is free of magnetic fields. Energy considerations show that the Meissner effect gives rise to a repulsive force between a permanent magnet and a superconducting sample. Meissner effect phenomena in YBa2Cu3Oz superconductors are quite complicated because a strong magnetic field can penetrate small regions of the sample by forcing these regions to go normal. These regions are in the form of tubes, with magnetic flux concentrated along their axes. Materials in which large numbers of normal regions can coexist with superconducting regions are designated type II superconductors.3,4 This is to be contrasted with type I metallic superconductors where the entire sample remains superconducting until the critical field is exceeded and the entire specimen goes normal. Almost all pure metals are type I superconductors, but there are exceptions, such as Nb and V.
(a) Preparation of the sample material. Our current recipe here at Middlebury calls for amounts of yttrium oxide Y2O3, barium peroxide BaO2, and copper II oxide CuO which would result in a mixture with formula YBa2Cu3O8.5. Determine the mass of each component needed to make 10.0 g of this formula. Obtain the correct amount of each oxide and then place the three powders in the 4 in diameter agate mortar. Use care when handling the chemicals and the samples. In particular, avoid breathing the powders and wash your hands after working with these materials. Mix and grind the mixture until it appears homogeneous.
Locate the Spex 30 ton press and the 1.25 in diameter die that should be sitting on the table nearby. Take the die apart and clean each piece. With its shiny side up, place one of the small end-pieces on the center of the base. Snap the cylindrical wall of the die over the base. Load the disk die with your mixture, spreading it into an even layer. Place the small top end-piece on top of your sample with the smooth surface toward the powder. Insert the piston into the die and place the die assembly in the center of the pressure plate of the press. Making sure that the valve on the right hand side of the press is closed, pump the pressure up to 20 ton/in2. Allow the sample to remain under pressure for about 15 minutes. Open the valve slowly. As soon as you can remove the die, return the valve to the closed position.
(b) Heating the sample. When you have removed your pressed sample from the die, take it to the computer-controlled furnace. Place your sample near the rear of the furnace on a platinum wire form that will hold it about 1 cm from the furnace floor. Turn on the IBM PC computer that controls the furnace. Edit the run-time data so that all files created by the program will bear your name. Check the temperature program to make sure that the program will perform the following steps.
- (i) Hold a furnace temperature of 750 K for two hours. This should expel any moisture or carbon dioxide from the sample and the BaO2 should melt.
- (ii) Heat the sample to 1230 K and remain at that temperature for about 20 minutes. At this temperature, excess oxygen can escape and Ba mobility is increased.
- (iii) Cool the sample to room temperature over a period of several hours.
(c) Forming the final sample. Crush and grind your cooled sample with the agate mortar and pestle until you obtain a coarse powder. Press the sample into a disk just as you did in step (a). Your sample is now ready for testing.
(d) Test for the Meissner Effect. Fill the large-mouthed dewar with liquid nitrogen. Place your sample on the stage and elevate the dewar until the sample is wet. Wait until the boiling subsides. Lower a tiny magnet on a string to a point near your sample. Test to see if the sample and the magnet repel each other. Note the approximate angle of the string when the gap between the sample and the magnet is just visible. If there is sufficient repulsion, try floating the magnet above the sample.
(1) Loss of electrical resistivity in Sn metal near 3.7 K.
Use your temperature and resistivity data for the Sn wire sample to prepare figures similar to Figs. 1 and 2. Be sure that you obtain a copy of the calibration sheet for the Ge resistance thermometer so that you can convert microvoltmeter readings to temperature.
(a) From your data, what is the transition temperature Tc for the Sn wire sample? What is the uncertainty in your determination of Tc? How well does your value agree with the value Tc = 3.722±0.001 K given in Ref. 8?
(b) Using Eq. (1) and your data, estimate HO for your Sn sample. What is the uncertainty in your determination of HO ? How well does your value agree with the value HO = 305±2 G given in Ref. 8?
(2) Meissner effect in YBa2Cu3O7 at 77 K .
(a) Give a detailed explanation for why a permanent magnet levitates above the surface of a superconductor. You may wish to review your knowledge of energy storage in magnetic fields and the relationship between forces and the gradient of potential energy functions. To help support your argument, draw an appropriate diagram showing the magnetic field lines in the region between the permanent magnet and the superconductor surface.
(b) When the permanent magnet is put directly on the surface of the superconductor, and then released, the permanent magnet does not rise to a levitating position. Why doesn't levitation take place? Read Ref. 1 thoroughly before answering this question. Explain why this might be evidence that your sample is a type II superconductor.
1. R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (John Wiley & Sons, New York, 1985), pp. 484-492.
2. P.A. Tipler, Modern Physics (Worth Publishers, New York, 1978), pp. 345-347.
3. C. Kittel, Introduction to Solid State Physics, 5th ed. (John Wiley & Sons, New York, 1976), pp. 357-398.
4. A.C. Rose-Innes and E.H. Rhoderick, Introduction to Superconductivity, 2nd ed. (Pergamon Press, New York, 1978), p. 7.
5. A.C. Rose-Innes and E.H. Rhoderick, ibid., pp. 16-20.
6. K.E. Tellier, private communication.
7. S.B. Qadri, L.E. Toth, M. Osofsky, S. Lawrence, D.U. Gubser, and S.A. Wolf, Phys. Rev. B35, 7235,1987.
8. Handbook of Chemistry and Physics, edited by R.C. Weast, 63rd ed. (CRC Press, Boca Raton, FL, 1983), p. E-85.
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