Dresner, Stability of superconductors.2002

as the temperature approaches absolute zero. One school of thought held that the electrons would freeze and stop moving, which implied an infinite resistance. Another school, using thermodynamic reasoning based on Nernst’s theorem, argued that the resistance would approach zero smoothly with falling temperature.

As Mendelssohn notes, in 1911 Kammerlingh-Onnes already knew that neither of these ideas applied categorically: early experiments on platinum and gold had showed that at low enough temperatures the resistance approached a finite limit and became independent of temperature. This limit is called the residual resistance and plays an important role in practical matters of magnet design. KammerlinghOnnes rightly attributed the residual resistivity to the scattering of conduction electrons by impurity atoms. What then, he asked, would happen in a very pure material?

Mercury was the clear choice for a test metal because it could easily be purified by multiple distillation. When Kammerlingh-Onnes measured its resistance as a function of temperature, he found that it dropped suddenly at 4.15 K to unmeasurably low values (< 10¯6 of the room temperature value). Such a sudden drop in resistance came as a great surprise, and when it was found to happen also in other metals (e.g., Pb, Sn, In), it inflamed Kammerlingh-Onnes’s mind with visions of resistanceless magnets producing high magnetic fields. Certain difficulties unforeseen (and indeed unforeseeable) dashed these early hopes for high-field magnets, but in the end they were realized, as described below.

In the meanwhile, a question of principle remained: How small was the resistance on the low-temperature side of the discontinuity? Now, measurement alone can never show that a physical quantity is zero, because of the finite precision of our apparatus. Today, based on theories created to explain what we know of superconductivity, we believe the resistance in the superconducting state to be exactly zero. But Kammerlingh-Onnes, with only experiment as his guide, could not be sure.

Accordingly, he set about improving his value of the upper bound to the resistance. He fashioned a loop of superconductor, charged it with current, and monitored the magnetic field outside the dewar flask for any decay. During several hours, he saw none, which bespoke a resistance < 10¯11 of the resistance at room temperature. Mendelssohn (1968) mentions such a persistent-mode experiment of much later vintage that lasted for two years and came to an end only because a transport strike interrupted the supply of liquid helium.

These early researches, which gave birth to the science of superconductivity, were honored in 1913 by the award to Kammerlingh-Onnes of the Nobel Prize in Physics.

CHAPTER 1

Table 1.1. Critical Fields and Temperatures of Some

Superconductors

the intercept on the field axis Bc is called the critical field. Table 1.1 gives the critical fields and temperatures of several of the superconductors discovered in the early days of superconductor research.

When a field Bp is applied to the superconductor, it remains superconducting up to some temperature Tp < Tc. The pair of values (Tp, Bp) are the coordinates of some point on the curve separating the superconducting from the normal state. A temperature like Tp lying at an interior point of the curve is often loosely referred to as the critical temperature; a clearer but clumsier nomenclature would be the critical temperature at a field Bp. But in most of the literature, the reader is left to decide from the context what temperature is meant by the designation critical.

After Kammerlingh-Onnes’s initial discoveries, superconductivity research lay fallow for more than two decades owing to the low values of the critical fields noted in Table 1.1. Then Meissner and Ochsenfeld (1933) discovered that superconductors expelled the applied magnetic field from their interiors. They found that if a material was cooled into the superconducting state and then exposed to an external magnetic field, the field would not penetrate the interior of the material until the field reached the critical value (at the particular temperature). Furthermore, they found that if the external field was first applied and then the temperature was decreased, when the temperature dropped below the critical value (at that field), the sample, as it became superconducting, expelled the field from its interior. In short, superconductors are not just perfect conductors, they are also perfect diamagnets. This phenomenon is called the Meissner effect.

1.3 TYPE-II SUPERCONDUCTORS

At first, no one knew that there were two kinds of superconductors. The superconductors discovered by Kammerlingh-Onnes and described above are now known as type-I superconductors or soft superconductors. Today, we know that there is another class of superconductors, known as type-II superconductors or hard superconductors that differ from the type-I superconductors in that they admit the magnetic field into their interiors while still remaining superconducting. It is from

Figure 1.2. The phase diagram of type-II superconductors.The lower left-hand region represents a superconducting state exactly like that of type-I superconductors: zero resistance and perfect diamagnetism. The upper right-hand region represents the normal state. The middle region represents the mixed state. The fields Bc1 and B c2 are called the lower and upper critical fields, respectively.

these type-II superconductors that contemporary scientific and commercial superconducting magnets are wound.

Shown in Fig. 1.2 is a phase diagram for the type-II superconductors similar to the phase diagram in Fig. 1.1. But now instead of two regions in the field-tem- perature plane, there are three. The lower left-hand region represents a superconducting state exactly like that of type-I superconductors: zero resistance and perfect diamagnetism. The upper right-hand region represents the normal state. The middle region represents a state, called the mixed state, that requires some detailed description.

The magnetic field that a type-II superconductor admits in the mixed state is not uniform, as classical physics requires, but is confined to discrete bundles called fluxoids (see Fig. 1.3). Each bundle has a normal (i.e., nonsuperconducting) core that is threaded by magnetic field. Outside the core, where the material is superconducting, the magnetic field drops off exponentially. The total magnetic flux

constant (6.625 x 10¯34 J s) and e is the charge on the electron (1.602 x 10¯19 C). Circling each core in the superconducting region, where the field is falling off, is an electric current, whose strength also falls off exponentially with distance from the core.

The existence of the triangular flux lattice shown in Fig. 1.3 was first proposed on theoretical grounds by Abrikosov (1957). Then Essmann and Träuble (1967) visualized Abrikosov’s flux lattice directly. They exposed the surface of a type-II

Figure 1.3. The triangular fluxoid latticeof a type-II superconductor in the mixed state. Each fluxoid has a normal core that is threaded by a quantum of flux. An electric current circles each core in the superconducting region.

superconductor to a puff of cobalt vapor produced by exploding a cobalt wire in the same (cold) chamber as the sample. The cobalt atoms, being ferromagnetic, followed the field lines in their outward flight from the wire and converged on the normal cores where the cores ended on the sample surface, revealing with great clarity the triangular lattice predicted by Abrikosov.

If a current is established in the sample perpendicular to the applied magnetic field (Fig. 1.4), it flows in the superconducting region between the normal cores. This current, called the transport current, by flowing through a region in which there is a magnetic field, feels a sidewise volume force J x B, where J is the transport current density and B is the local field. By the equality of action and reaction, the flux lattice feels a volume force -Jx B. (At this point, it is worth stating explicitly that the flux lattice is not just the triangular array of normal cores, but the entire continuous array of normal cores, circulating currents, and inhomogeneous magnetic field that fills the whole superconductor.) This volume force causes the flux lattice to move as indicated in Fig. 1.4. As fluxoids reach one surface of the sample, they disappear, while new fluxoids appear at the opposite surface, so that a steady flow through the sample is maintained.

If one moves with a velocity v through a region of magnetic field B, one feels an electric field E = v x B. Now the velocity of the transport current relative to the flux lattice, is parallel to J x B, so that in the rest frame of the transport current there is an electric field E that has the direction (J x B) x B. The electric field E thus always has a component in the direction opposite to that of J (see Fig. 1.5). So the

unable to move. In such a case, the flux-flow resistance vanishes, and the transport current, once established, persists just as in a type-I superconductor.

1.4. PINNING

The immobilizing of the flux lattice by defects and impurities is called pinning. If a transport current (current density J) is flowing in the superconductor and if it is exposed to a magnetic field B transverse to the current (as is typically the case in a conductor in a solenoid), the volumetric Lorentz force JB acting between the current and the fluxoids is restrained by the pinning. But there is a limit to how great a force the pinning centers can sustain, and if the product JB becomes too large, the flux lattice is torn loose and begins to move. Flux-flow resistance appears, and for practical purposes the superconducting ideal of lossless current flow is vitiated.

In a fixed background field B, the largest current density that a type-II superconductor can sustain without the appearance of flux-flow resistance is called the critical current density Jc. The rise in resistivity in a superconductor when the critical current density is surpassed is very rapid, and almost immediately beyond the critical current density, the flux-flow resistivity exceeds by a large margin the residual resistivity of ordinary conductors like copper. So the critical current density can be looked upon as a kind of boundary between two states, one superconducting and the other resistive. In this respect there is a loose analogy between the type-I and type-II superconductors. But it should be emphasized that there is in fact an important difference. When type-I superconductors first become resistive, the superconducting state has been destroyed, but when type-II superconductors first become resistive the superconducting state of quantized fluxoids still persists .

Experiments, among which the earliest were those of Kim, Hempstead, and Strnad (1963), show that the pinning force JcB can be considered independent of B to a rather good approximation. These authors have proposed on the basis of their experiments the empirical formula

where Bo is a constant of the order of a few hundredths of a tesla. At high fields it can be neglected and the critical current density taken inversely proportional to the field.

Not only does the critical current density Jc decrease with increasing field, but it also decreases with increasing temperature. This is because the pinning energy decreases with increasing temperature (cf. Eq. (2.4.8)). The higher the temperature, therefore, the smaller the Lorentz force JcB required to tear the entire lattice loose and initiate steady flux flow.

Even when the pinning force is strong, the flux lattice can creep due to thermal fluctuations. Since the pinning energy is comparable with the thermal energy kT, where k is Boltzmann’s constant and T is the temperature, thermal fluctuations occasionally lift a pinned fluxoid from its pinning site, after which it drifts under the action of the Lorentz force until it encounters another site. Such flux creep is accompanied by the appearance of a small flux-flow resistivity.

Figure 1.6 shows a surface in a three-dimensional space whose axes are temperature T, external magnetic field B, and current density J. Points below the surface correspond to a firmly pinned fluxoid lattice and the absence of resistance to current flow. Points above the surface correspond to flux flow and the presence of resistance. Inside the larger surface, near the origin, is a second much smaller surface that marks the boundary of type-I behavior. At points below that surface, the superconductor exhibits a Meissner effect and is a type-I superconductor. The bounding curve of the smaller surface in the B-T plane is the same as the phase boundary in Fig. 1.2 that separates the Meissner from the mixed state. The current on this lower surface is determined by the Silsbee condition that the sum of the self-field and the background field always be less than the critical field that separates the Meissner from the mixed state (Silsbee, 1916).

On the other hand, the current on the upper surface is determined by the condition that the flux lattice remain pinned, i.e., that there be no flux-flow resistance. The bounding curve of this surface in the B-T plane need not necessarily be the phase boundary in Fig. 1.2 that separates the mixed state from the normal state, and in fact for some of the new high-temperature ceramic superconductors (Tc ~ 100 K ) it is not. For some of those superconductors, there is yet another curve, called the irreversibility curve, that traverses the region of the B-T plane corresponding to the mixed state: on one side of it Jc is finite, on the other Jc = 0. It is the irreversibility curve that determines the practical performance of the high-tem- perature superconductors rather than the phase boundary between the mixed and normal states. On the other hand, for the older, low-temperature superconducting metallic alloys and compounds (e.g., NbTi and Nb3Sn; Tc = 9.6 K and 18 K, respectively), the irreversibility curve and the phase boundary are practically indistinguishable and usually no distinction is made between them. The difference in behavior between the two classes of superconductors is caused, of course, by the quite large difference in the amplitude of the thermal agitation of their fluxoid lattices.

The surface in Fig. 1.6, the so-called critical surface, is the basic datum on which the considerations of this monograph are based. With it in hand we can turn to the stability problem; but first we must begin with a short description of the structure of modern superconducting wires.