Equipment Used:

yttrium barium copper oxide ring made by Colorado Superconductor, Inc., Applied Magnetics GM1A gaussmeter with PT-72 probe

Objective:

to record the magnetic field of a current induced in a ring of high-T_c superconductor, and to observe the persistence of the current due to negligible resistance of the superconducting material.

This experiment is based on an article by Liu, Tucker, and Heller, Am. J. Phys. 58 (3) March 1990, 211-218.

Superconductivity

A superconductor is a perfect conductor, having zero resistance, and it is also a "perfect diamagnet": weak magnetic fields may not pass through the body of the superconducting material. This latter effect is known asthe Meissner effect and is the basis of the now-familiar levitation demonstrations using superconducting material. If a magnetic field is applied to the material while it is superconducting, the applied field must be canceled out within the material by the field of a current which is induced near the surface of the material. The current distribution assures that while field does actually penetrate into the superconductor, it dies away exponentially with depth into the material; the characteristic decay length is l_L, the London penetration depth, which is of the order of a few times 10^-8 m.

Strictly speaking the previous comments about exclusion of field from a superconductor apply to Type I superconductors, for which l_L is small compared to the coherence length x -10^-8 m, another characteristic length which measures the distance required for transition from normal to superconducting state. If l_L is larger than x, there results a Type II superconductor, in which a sufficiently large field can penetrate to form a lattice of thin tubes of normal, field-carrying material threading the superconducting material.

Origin Of Superconductivity

All superconductors discovered between 1911, when Kamerlingh Onnes first observed superconductivity in mercury at 4.15 K, and 1986 can be understood in terms of the 1957 theory of Bardeen, Cooper, and Schrieffer (BCS theory). According to this theory there is an attraction between electrons in a superconductor which leads to a collective state of the material that has a particularly low energy, separated from the higher-energy "normal" (resistive) states by an energy gap on the order of 10^-4 eV. This is to be contrasted with the energy gap that occurs between the valence band and the conduction band in a semiconductor; that gap is of the order of 1 eV and is due to the interaction of electrons with the periodic lattice of atoms. In BCS theory the mechanism of attraction between superconducting electrons is well understood: an electron interacts with the crystal lattice and deforms it, and another electron responds to the deformed lattice; thus the electrons interact through the lattice deformation. Lattice deformations are quantized as phonons, so we say that a pair of electrons is coupled via a phonon. The coherence length x is also a measure of the range over which the electrons interact to form a pair. There's more: the pair couples so that the spins of the electrons are opposite ("singlet" pair, S = 0) , and also their linear momenta are opposite (pair momentum = 0) . In the superconducting state, all the electrons are paired, and the state of the whole system consists of a superposition of these pairs.

In 1983 the highest temperature T_c known for a transition from normal to superconducting state was 23.2 K for Nb₃Ge. Georg Bednorz and Alex Muller began a search for new materials with very strong electron-phonon interaction, with the aim of finding materials which become superconducting at even higher temperatures. In 1986 they succeeded, finding a copper oxide ceramic with T_c - 30 K. This watershed work earned them the 1987 Nobel Prize, and it led to an explosion of research which within two years produced other copper oxide ceramics with T_c up to 125 K. These "high-T_c" superconductors are Type II, and because they become superconducting above the boiling point of inexpensive liquid nitrogen (77 K), there are high hopes for technological applications.

What is the origin of high-T_c superconductivity? Despite intense research effort since 1986, the answer is still not known. Ineach of the high-T_c superconductors the superconductivity is found in planes of copper and oxygen atoms; the spacing between the planes can exceed the very short coherence length x ~ 10^-9 m. Experiments show that the superconductivity is due to S = 0 pairs of electrons, but it is not clear what the coupling mechanism is. Some think that phonons are again the mechanism, but the BCS theory is not sufficient to explain the high-T_c. superconductors.

Magnetic Field Of A Current In A Superconducting Ring

From the law of Biot and Savart one can derive the magnetic field on the axis of a thin current-carrying wire ring. It is

where r is the radius of the ring and z is the distance, measured along the axis, from its center. Without restricting to the axis, one can see that the field of the ring takes the form of a magnetic dipole field, with its orientation depending on the direction of the current flow.

Consider a ring of material which is "normal" at room temperature but becomes superconducting at low temperature T_c. Suppose a magnetic field is externally applied along the axis of the ring when it is normal. When the ring is cooled below T_c the applied field continues to poke through the hole in the ring, but currents flow to offset the applied field within the superconducting material. If the externally applied field is removed, these currents are no longer needed and stop flowing. But is that the whole story? No, Lenz's Law assures us that Nature wants no change in the flux through the ring. When the externally applied field is removed, there must be induced a new current that maintains the flux through the hole in the ring. If the resistance of the superconductor is truly zero, once this current is induced, it persists indefinitely!

Satisfy yourself that if the applied field B is initially upward through the superconducting ring, when it is removed there is induced an upward field B due to a current flowing in the ring, in the direction counterclockwise seen from above. Turn the ring over, and the current persists in the same direction so that its field is now downward!

Now suppose that the ring is broad, so that the difference in inner radius r₁, and outer radius r₂ is comparable to r_l. An expression for the magnitude of the field B on the axis may be obtained by adding up the contributions of concentric currents dI distributed uniformly on the top and bottom surfaces of the ring. Alternatively, we may use the expression already given f or a "thin" wire loop, with r now being replaced by an "effective radius" r_e of the broad ring:

We see that a plot of B^-2/3 versus z² is a straight line with intercept b and slope m, so that

and from the value of B measured at z, one can calculate the effective ring current I.

Notice that we are making the assumption that the current flows in concentric circles around the center of the ring. It is conceivable that the field on axis could arise from a distribution of currents circulating in the proper sense but not centered on the center of the ring. Liu, Tucker, and Heller show that one such distribution, a set of "whorls" equivalent to thin current loops oppositely directed at r₁ and r₂, gives rise to a z dependence different from that in the above equation, so that observing B(z) like that in the equation is support for the assumption of concentric currents.

Y123 Ring

You are provided with a commercially made ring of YBa₂Cu₃O_7-? which has r₁ = 0.60 cm, r₂ = 1.50 cm, and thickness 0.63 cm. The dimensions are determined by convenience in fabrication from the brittle material. This material, referred to informally as Y123, has superconducting transition temperature of about 90 K, well above liquid nitrogen temperature.

If the resistance of the superconducting Y123 were really zero, an induced current would persist indefinitely. Actually one reason that the high- T_c superconductors have been somewhat disappointing for technological applications is that they do demonstrate some resistance, which has been attributed to "flux creep", with dissipation of energy resulting as magnetic flux lines are dragged through the material. Liu, Tucker, and Heller report observing exponential decay of the field due to the current induced in the ring; they associate this with a t(=L/R) time constant:

where L is modeled for a thin ring of radius

,                                                                      (5)

with B from (1) for z = 0. This gives a lower limit on L as a few nano-henries, and from t and L they obtain an upper limit estimate of R.
 
 

For Your Report

1) Observe the field B due to persistent current in a Y123 ring. You can induce this current by cooling the ring through T_c in an applied magnetic field, then removing that applied field while the ring remains below T_c. Satisfy yourself that the direction of the field due to the persistent current is as expected. Turn the ring over. If the ring contains a single continuous loop of current, turning the ring over should simply reverse the direction of its field B. If the induced current consists of a collection of little loops, none of which goes all the way around the ring, the result may not be so simple; in this case, the measured field could vary around the circumference of the ring. What do you observe?

2) Measure the axial B with the ring as close as possible above the Hall probe. What is the smallest z value accessible? Compensate for gaussmeter drift by taking the difference of readings with the ring above the probe and with the ring far away. Work quickly; the ring must not be allowed to warm up and go normal!

(NOTE:  1 gauss = 10^-4 T.)

3) Record the axial B f or several z values, obtained by inserting nonmagnetic spacers. Compare to the axial field of a "thin" loop (2); this is conveniently done with a spreadsheet. Fit a straight line to your values of B^-2/3 versus Z², and so evaluate r_e and I with this model.

4) We are going to observe persistence of the induced current by recording the axial field B over the next few weeks. An aluminum "basket" and storage dewar are provided for holding the ring at liquid nitrogen temperature. At each lab period, we'll take the average of 10 readings of the field B beneath the center of the ring, with the ring in the "basket" and placed as close as possible to the gaussmeter probe. We'll keep this up as long as luck and perseverance last. (In 1991 we got tired of following the decaying field after 2 months.) The results of measurements in all lab periods will be compiled and distributed. Look for an exponential decay of the field B, and if you observe it, attribute it to an L/R time constant and estimate an upper limit on the resistance R of the ring.