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2.6.4 Superconductivity

Superconducting properties of materials

Superconductivity is characterised by the complete loss of electrical resistance below some finite temperature, T_c, which is a characteristic of the material used. The phenomenon is rather common: one quarter of the elements become superconducting, as well as several thousand different alloys and compounds. T_c values vary from 0K to, at the time of writing, 160K. Above T_c superconductors exhibit ohmic conductivity, much as for a normal metal. At and below T_c, in the absence of a magnetic field, the low frequency resistivity falls to zero. Superconductivity can be destroyed by the application of a magnetic field which exceeds a critical value H_c(T), a decreasing function of temperature T, which reaches zero at T = T_c. The critical fields of superconductors at T = 0 vary widely, having rather low values for pure elemental superconductors (typically 10 mT) but attaining very high values, of order 100 T, for the new high temperature ceramic materials (see below).

(Click the Images below to view Larger Images)

Superconductors fall into two classes: Type I and Type II. The principal difference between them is that the former possess a single well defined critical field whereas the latter have two critical fields, the lower of which (H_c1) defines the field below which magnetic flux is substantially excluded from the material (the Meissner effect). For fields above H_c1 but below the upper critical field (H_c2) the superconductor is said to be in the mixed state and magnetic flux penetrates it although a supercurrent can still flow without dissipation. In the mixed state any penetrating magnetic flux is confined within narrow flux vortex lines with each of which is associated a quantum of magnetic flux Φ₀ ( = h/2e = 2 x 10⁻¹⁵ Wb). The two figures (left) show the variation of H_c1 with temperature for a number of elements (which are mainly Type I superconductors) and the temperature variation of H_c2 for a number of alloys. (B_c1 =  μ₀H_c1 and B_c2 = μ₀H_c2.)

The critical current I_c of a Type I superconductor is that current which will produce the critical magnetic field at the conductor’s surface. The passage of a current down a Type II superconducting wire in the mixed state exerts a Lorentz force on the magnetic flux vortices, unless these lie exactly parallel to the current flow. If the vortices are caused to move by this force then dissipation will occur and appear as electrical resistance. To make conductors with high I_c values (essential for construction of high field solenoids etc.) Type II materials are designed so that the microscopic structure favours ‘pinning’ of flux vortices at fixed positions in the material, even in the presence of a large Lorentz force. This requires great attention to be paid to structural morphology but is readily achieved in a number of alloys based on niobium. The ability to produce critical current densities as high as 10¹⁰ A/m² has been demonstrated routinely in helium cooled superconductors and in thin films of the high temperature cuprate materials (see below).

The resistance of superconductors to alternating currents is not strictly zero, due to the inertia of the paired charge carriers which are responsible for superconductivity in all presently known materials. The resistivity nevertheless can be extremely small at all temperatures below T_c. The real conductivity of normal metals means that the surface resistance R_s increases as f ^0.5, where f is the frequency of the applied current. In contrast, the complex conductivity of superconductors leads to an f ² dependence. For high purity conventional superconductors in single crystal form the surface resistance at temperatures well below T_c has the following dependence

   R_s (f²/T) exp(−Δ(T)/kT)

where Δ(T) is the temperature dependent energy gap between the superconductor ground state and the normal conducting state. For conventional superconductors the energy gap is of order 1 meV and its value at T = 0, Δ(0) scales with T_c. Δ(T) is a decreasing function of T reaching zero at T = T_c. The above equation applies for frequencies f << Δ/h and shows that for T/T_c < 0.2 the surface resistance of pure superconductors at microwave frequencies is many orders of magnitude lower than for conventional metals at the same temperature. Also the surface reactance X_s is to a good approximation proportional to f. This may be seen as a consequence of the frequency independent penetration depth λ(T) of the magnetic field.

Properties of superconducting elements

Recent developments in superconductivity

Since 1987 the number and variety of known superconductors has enlarged very significantly. The most important development has been the observation of superconductivity in a wide range of ternary or quaternary cuprate compounds. The original discovery of a T_c as high as 40 K in the LaSrCuO system by Bednorz and Muller almost doubled the record highest transition temperature. Further discoveries have resulted in a number of families of related compounds which are demonstrating potential for real applications and possessing T_c as high as 133 K at normal pressure. The cuprate superconductors are ternary, quaternary or even more complex oxides of copper in which planes of copper and oxygen are separated from one another by the other cations. An example of the structure of the unit cell of one of the most promising compounds for commercial applications YBa₂Cu₃O₇ is shown in the figure below. Although the 2-dimensional planes are responsible for superconductivity the materials carry a supercurrent in three dimensions, adjacent copper-oxygen planes being coupled together sufficiently strongly to allow resistanceless current flow. The cuprate superconductors are all anisotropic to a greater or lesser degree however. This anisotropy is observed in penetration depth λ(0), coherence length, critical fields and critical current densities.

In addition to the cuprates the transition temperature of organic superconductors has been raised to 11K and it has also been shown that doping of Buckminsterfullerene (a ‘soccer-ball’ shaped molecular form of carbon) with alkali metals can produce superconductivity at 40 K. Both of these latter developments have been somewhat eclipsed by the technological importance of the cuprates. A further class of metallic superconductors, called the heavy fermions, has also emerged recently. These superconduct mainly in the temperature region below 1 K. The interest in them has been mainly generated by the novel mechanism believed to be responsible for the charge carrier pairing and in the symmetry of the resulting macroscopic wave function. In all four cases of ‘exotic superconductivity’ mentioned above, the precise mechanism for superconductivity is as yet not clear though all involve paired charge carriers. It seems at the time of writing that the BCS mechanism responsible for superconductivity in the elemental metals, in which electron pairs couple through exchange of virtual phonons, does not provide the only or even the dominant mechanism in the high temperature cuprates superconductors.

The table below lists T_c, H_c2 (at a specified temperature) and λ(0) for a selection of alloy and compound superconductors. In general where a range of stoichiometries exhibit superconductivity the composition with the highest reported T_c has been given. Note that the list represents only a small selection of the total of known superconductors.

Properties of some superconducting alloys and compounds

Values of T_c, H_c2 and λ are to some extent dependent on composition and method of preparation. The above values represent only a guide to the optimal values to be expected. Where an entry is blank the parameter in question has not been measured or a number of measurements are discrepant.

J.C.Gallop

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