4.3 Type-II superconductors
For decades it was assumed that all superconductors, elements and alloys, behaved in similar ways, and that any differences could be attributed to impurities or defects in the materials. However, in 1957, Abrikosov predicted the existence of a different sort of superconductor, and Figure 23 shows direct evidence for the existence of what are now known as type-II superconductors. A comparison of Figures 23 and 22 indicates that the effect of an applied field on a type-II superconductor is rather different from that for type-I superconductors.
For simplicity, we shall consider first a long cylindrical specimen of type-II material, and apply a field parallel to its axis. Below a certain critical field strength, known as the lower critical field strength and denoted by the symbol Bc1, the applied magnetic field is excluded from the bulk of the material, penetrating into only a thin layer at the surface, just as for type-I materials. But above Bc1, the material does not make a sudden transition to the normal state. Instead, very thin cylindrical regions of normal material appear, passing through the specimen parallel to its axis. We shall refer to such a normal region as a normal core. The normal cores are arranged on a triangular lattice, as shown in Figure 23, and as the applied field is increased, more normal cores appear and they become more and more closely packed together. Eventually, a second critical field strength, the upper critical field strength Bc2, is reached, above which the material reverts to the normal state. The state that exists between the lower and upper critical field strengths, in which a type-II superconductor is threaded by normal cores, is known as the mixed state. As Figure 24 shows, both the upper and lower critical field strengths depend on temperature in a similar way to the critical field strength for a type-I material (Figure 11).
The normal cores that exist in type-II superconductors in the mixed state are not sharply delineated. Figure 25 shows how the number density of superelectrons and the magnetic field strength vary along a line passing through the axes of three neighbouring cores. The value of ns is zero at the centres of the cores and rises over a characteristic distance ξ, the coherence length. The magnetic field associated with each normal core is spread over a region with a diameter of 2λ, and each normal core is surrounded by a vortex of circulating current.
You can see from Figure 25 that the coherence length ξ, the characteristic distance for changes in ns, is shorter than the penetration depth λ, the characteristic distance for changes in the magnetic field in a superconductor. This is generally true for type-II superconductors, whereas for type-I superconductors, ξ > λ (Figure 20). For a pure type-I superconductor, typical values of the characteristic lengths are ξ ~ 1 μm and λ = 50 nm. Contrast this with the values for a widely-used type-II alloy of niobium and tin, Nb3Sn, for which ξ ~ 3.5 nm and λ = 80 nm.
The reason that the relative magnitude of the coherence length and the penetration depth is so important is that when ξ > λ, the surface energy associated with the boundary between superconducting and normal regions is positive, whereas when ξ < λ, this surface energy is negative. Justifying this statement would involve a discussion of the thermodynamics of superconductors, but for our purposes it is sufficient to just look at the consequences. For a positive surface energy, the system will prefer few boundaries and we expect relatively thick normal and superconducting regions, as observed in the intermediate state in type-I materials. Conversely, a negative surface energy favours formation of as much boundary between normal and superconducting regions as possible, and this is what happens in the mixed state in type-II materials with many narrow normal cores. The lower limit for the diameter of the cores is 2ξ, as shown in Figure 25, because ξ is the distance over which ns can vary.
This energy argument does not explicitly indicate how much magnetic flux passes through each of the normal cores. However, quantum mechanical arguments show that the magnetic flux linking any superconducting loop must be quantised, and that the quantum of magnetic flux is
h / 2e = 2.07 × 10−15T m2
where h = 6.63 × 10−34J s is Planck's constant. In fact, each of the normal cores shown in Figure 23 contains just one quantum of flux, since this is more favorable energetically than having two or more quanta of flux in a core.
The quantisation of flux in a superconductor is of particular importance in SQUIDs, the superconducting quantum interference devices that are at the heart of the magnetometers used for measuring the magnetic fields produced by currents in the brain. A SQUID contains a small loop of superconductor with a weakly superconducting link in it, and the quantisation of flux in the loop causes its electrical properties to depend on the flux applied to it, with a periodicity equal to the flux quantum. The very small magnitude of the flux quantum is responsible for the sensitivity of the device to very small magnetic fields.
A final point is worth noting about the quantum of flux: the factor of 2 in the denominator of the expression h / 2e is a consequence of the coupling of pairs of electrons in a superconductor and their condensation into a superconducting ground state. There is charge −2e associated with each of these electron pairs.
Critical currents in type-II superconductors
The high values of the upper critical field strength Bc2 of many type-II superconducting alloys make them very attractive for winding coils for generating high magnetic fields. For example, alloys of niobium and titanium (NbTi2) and of niobium and tin (Nb3Sn) have values of Bc2 of about 10 T and 20 T, respectively, compared with 0.08 T for lead, a type-I superconductor. However, for type-II materials to be usable for this purpose, they must also have high critical currents at high field strengths, and this requires some help from metallurgists to overcome a significant problem.
This problem is related to the interaction between the current flowing through a type-II superconductor in the mixed state and the ‘tubes’ of magnetic flux that thread through the normal cores. The electrons will experience a Lorentz force, perpendicular to both the current density and the magnetic field. We can regard this as a mutual interaction between the electrons and the flux in the normal cores, as a result of which each normal core experiences a force that is in the opposite direction to the Lorentz force on the electrons. The directions of the magnetic field, current and forces are shown in Figure 26.
This Lorentz force can cause the cores and their associated magnetic flux to move, and the flux motion will induce an emf that drives a current through the normal cores, somewhat like an eddy current. Energy is therefore dissipated in the normal cores, and this energy must come from the power supply. The energy dissipation means that the flow of electrons is impeded, and therefore there is a resistance to the flow of the current.
Flux motion is therefore undesirable in type-II superconductors, and the aim of the metallurgists who develop processes for manufacturing wire for magnets is to make flux motion as difficult as possible. This is done by introducing defects into the crystalline structure, particularly by preparing the material in such a way that it comprises many small crystalline grains with different orientations and small precipitates of different composition. These defects effectively pin the normal cores in position – they provide a potential barrier to motion of the cores, so that the force on the cores must exceed a certain value before the cores can move. The more of these flux pinning centres that are present, and the greater the potential barrier they provide, the greater will be the current required to set them in motion, i.e. the greater the critical current. So, unlike a normal conductor, for which improving the purity and reducing imperfections in the crystal structure lead to better conductivity, with type-II superconductors the inclusion of impurities and defects in the crystal structure can improve the critical current and make the material more suitable for use in electromagnets.
Undoubtedly the largest use of superconducting material for a single project is in the Large Hadron Collider at CERN, due for commissioning in 2007. The 27 km-long accelerator tunnel contains 1232 superconducting magnets that are responsible for steering the particle beams around their circular paths. Each of these magnets is 15 m long, has a mass of 35 tonnes and produces a magnetic field strength of 8.5 T. The coils in each magnet are made from about 6 km of niobium-titanium cable, with a mass of about a tonne, and will be maintained at a temperature of 1.9 K using liquid helium. To construct an accelerator with a similar specification using non-superconducting magnets would have required a 120 km tunnel and phenomenal amounts of power to operate.