# An Introduction to Superconductivity

Authored by: William F. “Joe” Vinen , Terry P. Orlando

# Handbook of Superconductivity

Print publication date:  July  2022
Online publication date:  July  2022

Print ISBN: 9781439817322
eBook ISBN: 9780429179181

10.1201/9780429179181-4

#### Abstract

Any material becomes more ordered as it cools, its entropy decreasing, provided it remains in thermodynamic equilibrium. As the material approaches the absolute zero of temperature, its entropy tends to vanish, so that it becomes completely ordered. Often the ordering proceeds through one or more discrete phase transitions. Perhaps the most remarkable of these phase transitions, and surely the most unexpected, are those leading to superfluid or superconducting phases, typically at temperatures of a few kelvin. At a superficial level, these phases can be seen to exhibit frictionless flow, but they have other striking macroscopic quantum properties that are a reflection of the unique and special type of quantum ordering that takes place within them.

#### A1.2.1  Superconductivity as an Ordering Transition

Any material becomes more ordered as it cools, its entropy decreasing, provided it remains in thermodynamic equilibrium. As the material approaches the absolute zero of temperature, its entropy tends to vanish, so that it becomes completely ordered. Often the ordering proceeds through one or more discrete phase transitions. Perhaps the most remarkable of these phase transitions, and surely the most unexpected, are those leading to superfluid or superconducting phases, typically at temperatures of a few kelvin. At a superficial level, these phases can be seen to exhibit frictionless flow, but they have other striking macroscopic quantum properties that are a reflection of the unique and special type of quantum ordering that takes place within them.

In the case of superconducting metals, the phase transition involves the electron fluid that is responsible for electrical conductivity in the normal (high-temperature) phase. The frictionless flow is seen as a loss of electrical resistivity, the material often showing a resistivity that is unmeasurably small. Electrical resistivity is due to the scattering of the conduction electrons by imperfections in the crystal lattice in which they are moving, so one might be tempted to think that in a superconductor these scattering processes are mysteriously turned off. As explained in Pippard's historical introduction, this view is quite misleading. All superconductors exhibit the Meissner effect: in sufficiently small applied magnetic fields, they behave as perfectly diamagnetic materials, in the sense that the magnetic flux is excluded in a reversible manner; they behave just like conventional diamagnetic materials, except that they have a much larger diamagnetic susceptibility. The required screening current is maintained as part of the equilibrium state of the system just as it is in the diamagnetic screening current in a diamagnetic atom or molecule. Scattering processes, far from having disappeared, are helping to maintain this equilibrium, as we shall see a little later.

The superconducting state is a new thermodynamic phase of the metal, distinct from the normal phase. The form of the heat capacity of the metal in the neighbourhood of the transition is very similar to that found in ordering transitions of the type seen in, for example, a paramagnetic material when it transforms into a ferromagnet. In the superconducting phase, the conduction electron fluid is in a more strongly ordered state than it is in the normal phase, and the diamagnetism observed in the Meissner effect is an equilibrium property of this ordered state.

Distinct phases can be conveniently described in an appropriate phase diagram: that for most pure metals (type I) is shown schematically in Figure A1.2.1, where the material has a shape with zero demagnetizing factor. The transition between the normal and superconducting phases in zero field is second order (no latent heat); in a finite field it is first order. The transition from superconducting to normal phase, as the field is increased at fixed temperature, takes place when the free energy associated with the induced magnetic moment exceeds the free energy difference (Fs Fn ) between the two phases in zero field, which leads to the following relationship for the critical field.

A1.2.1 $F n − F s = B c 2 / 2 μ 0$

Other and more complicated phase diagrams are also possible, as we shall see later.

Figure A1.2.1   Schematic phase diagram for a type I superconductor: applied magnetic flux density (B) plotted against temperature (T).

#### A1.2.2  The Meissner Effect and the Nature of the Ordering in a Superconductor

The simplest form of diamagnetism in a single atom arises when, as is often the case, the atomic electron wavefunctions ψ are not significantly perturbed by the applied magnetic field. The current electric density in the atom given in terms of the vector potential A for each electron acting independently by

A1.2.2 $J ( r ) = i e ℏ 2 m ( ψ * ∇ ψ − ψ ∇ ψ * ) − e 2 m ψ ψ * A ( r )$

then reduces to

A1.2.3 $J ( r ) = − e 2 m ψ ψ * A ( r )$

since the unperturbed wavefunctions make no contribution. The resulting diamagnetic moment is very small, because the atom is very small. Fritz London noticed that, if Equation (A1.2.3) was to apply to each conduction electron in a whole metal, the resulting diamagnetism would be very large, as in a superconductor, because the electron wavefunctions ψ extends over the very large volume of the whole metal. This does not happen in a normal metal because the conduction electron wavefunctions are not unperturbed; each electron wavefunction is strongly perturbed and describes a quantized cyclotron orbit. The two terms in Equation (A1.2.2) almost cancel, so that there remains only the very weak ‘Landau’ diamagnetism. But if the conduction electron wavefunctions were not modified by the field, so that Equation (A1.2.3) were to apply to a macroscopic number density (ns ) of electrons in the whole superconducting metal, the Meissner effect would be more or less correctly described. Equation (A1.2.3) would then imply that the total current in the superconductor would be given by

A1.2.4 $J = − n s e 2 m A$

from which we obtain, by taking the curl,

A1.2.5

where B is the magnetic induction. Combining Equation (A1.2.5) with the Maxwell equation curl B = μ0 J , we obtain

A1.2.6 $∇ 2 B = 1 λ L 2 B$

showing that the field tends to zero within the superconductor, with a penetration depth given by

A1.2.7 $λ L 2 = m μ 0 n s e 2 .$

If ns is of the order of the total number of conduction electrons per unit volume in the metal, λL is very small (of order 100 nm) so that flux exclusion is almost complete, as observed. Careful experiments show that a finite penetration depth does exist, with the order of magnitude given by Equation (A1.2.7), so we seem to have a good description of the superconducting behaviour.

The ‘rigidity’ in the wavefunction of the electrons in a superconductor that leads to the Meissner effect must presumably be a result of the ordering process that marks the onset of superconductivity. The nature of this ordering process became clear only when Bardeen, Cooper and Schrieffer had developed their theory, although the suggestion given much earlier by Fritz London that superfluidity in liquid helium and superconductivity in metals might have a common origin can now be seen to have pointed the way. Superfluidity is associated with Bose condensation: an ordering described by the accumulation of the helium atoms in a single quantum state. The BCS theory indicated, perhaps somewhat surprisingly, that a similar process is occurring in a superconductor. Since electrons are fermions, the individual electrons themselves cannot exhibit Bose condensation, but BCS told us that an attractive interaction between electrons, due to local distortion of the lattice (phonon exchange), leads to the formation of electron pairs (Cooper pairs), which can and do undergo a form of Bose condensation. A satisfactory description of this condensation is not straightforward, since each electron pair occupies a large volume (connected with the coherence length ξ0 to which we refer later), so that there is massive overlap between the pairs. Bose condensation leads to long-range order in a particular type of correlation function, which, in the case of liquid helium, is the single-particle density matrix. In the superconducting electrons, it is the two-particle density matrix that exhibits the same long-range behaviour, as can be shown directly from the BCS wavefunction.

It is the formation of the electron-pair condensate that gives rise to the rigidity of the superconducting wavefunction. Small perturbations to the wavefunction can be achieved only by mixing in states in which electron pairs have been removed from the condensate, and it turns out that this requires a minimum energy, Δ. This minimum energy is that required to produce thermal excitation of the system. In a normal metal, the excited states of the system are obtained by taking an electron from below the Fermi surface and placing it above, thus creating an electron excitation and a hole excitation. In the superconductor, the excitations involve the breaking of electron pairs, and they correspond to linear combinations of electron-like states and hole-like states in the normal metal. Each excitation has an energy equal to $( Δ 2 + ε k 2 ) 1 / 2$ , where εk is the energy of the corresponding excitation in the normal state. As the temperature of the superconductor is raised above absolute zero, more and more excitations are produced, and the energy gap Δ falls. Eventually, at the critical temperature, T c, Δ vanishes and the superconductor becomes a normal metal. In the BCS theory, T c is related to the energy gap at T = 0 by the relation Δ(0) = 1.76kBT c. However, it should be added that the existence of an energy gap is not essential for superconductivity; for example, superconductors containing magnetic impurities may be gapless; and superfluid helium is also gapless. The condensate in liquid helium owes its ‘rigidity’ to the form of the excitation spectrum, which is itself determined by the existence of the condensate.

Only the condensed electrons contribute to the Meissner effect, and the value of ns in Equation (A1.2.4) is the effective number of such electrons. The excitations behave to some extent like electrons in a normal metal. We are led, therefore, to a two-fluid model, in which ns electrons behave as superconducting and the rest as normal. The falling value of ns with increasing temperature is reflected in an increasing penetration depth (λL →∞ as TT c).

#### A1.2.3  The Ginsburg–Landau Equations; the Pippard and Ginsburg–Landau Coherence Lengths

It is convenient in developing an understanding of superconductivity to introduce the ‘condensate wavefunction’ or Ginsburg–Landau wavefunction. This can be formally defined in terms of the correlation function to which we have already referred, but in essence, it is a complex function, ψ , the phase of which is the phase of the wavefunction of the condensed pairs and the amplitude of which is proportional to the local concentration of condensed pairs. The function ψ obeys the Ginsburg–Landau equations when the temperature is near T c; at lower temperatures, these equations cease to be strictly correct, but they still provide a correct qualitative description. One of the G–L equations gives the supercurrent density.

A1.2.8 $J ( r ) = i e ℏ 2 m ( ψ * ∇ ψ − ψ ∇ ψ * ) − 2 e 2 m ψ ψ * A ( r )$

(cf Equation [A1.2.2]), while the other, which has the form of the nonlinear Schrodinger equation, describes in essence how the amplitude of ψ varies with position, such as might occur near a normal-superconducting boundary. It is found that ψ cannot change abruptly with position, but only gradually over a characteristic distance, ξGL, called the Ginsburg–Landau coherence length. We note the presence of the first term on the right-hand side of Equation (A1.2.8), which implies that current-carrying states of the superconductor are possible independently of the magnetic field. We discuss such states in a moment. Since the modulus of ψ is a measure of the density of condensed electrons, it also is a measure of the extent of the superconducting ordering. Indeed Ginsburg and Landau originally introduced ψ as a (complex) ‘order parameter,’ and it is frequently referred to in this way. Often it is referred to in other ways: ‘the gap parameter,’ since it is proportional to the energy gap in a spatially homogeneous situation; or the ‘pair potential,’ since it appears as a self-consistent potential in some formulations of the theory of superconductivity.

The superconducting wavefunction is not completely rigid in its response to a perturbation. It is rigid only for perturbations that vary slowly with position. The relevant characteristic length scale is the size of a Cooper pair ξ0, often called the Pippard coherence length. In practice, ξ0 is often larger than λ L, so that our derivation of λL as the penetration depth, which relied on the assumption of a rigid wavefunction, breaks down. In Equation (A1.2.4), the relation between the current density and the vector potential has to be replaced by a nonlocal relation, involving a kernel with range ξ0, and the expression for the penetration depth becomes more complicated (λ ≠ λL). For a pure metal, the coherence lengths ξGL and ξ0 are equal at low temperatures and equal to πΔ(0)/ћνF, where νF is the Fermi velocity in the normal metal, but at higher temperatures, they behave differently, ξ0 remaining constant, but ξGL diverging to infinity at T c.

#### A1.2.4  Persistent Currents and the Quantization of Trapped Flux

Suppose that the superconductor has the form of a long hollow cylinder, with radius R and wall thickness t. If t is large compared with the penetration depth, and if an external magnetic field is applied parallel to the axis of the cylinder, no magnetic flux penetrates to the inside of the cylinder. Suppose, however, that the cylinder is cooled into the superconducting phase while exposed to the external magnetic field. The state of lowest energy must surely then be one in which magnetic flux is trapped inside the cylinder. In such a state, the Ginsburg–Landau wavefunction must, as it turns out, have the form

A1.2.9 $ψ = ψ 0 exp [ i S ( r ) ]$

where the ψ 0 is the unperturbed wavefunction, and the phase S varies round the ring. The corresponding current density, obtained from Equation (A1.2.8), is

A1.2.10 $J = e ℏ ψ 0 2 m ∇ S − 2 e 2 ψ 0 2 m A .$

Taking the curl of this equation we recover Equations (A1.2.5) and (A1.2.6), so that the magnetic field is still screened from the bulk of the superconductor. However, there is now a magnetic field within the cylinder, as we see by taking the line integral of Equation (A1.2.10) round a circuit C enclosing the hole in the superconductor and lying at a distance much greater than the penetration depth from the surface (Figure A1.2.2, where we have assumed that on this circuit, and therefore, the flux within the circuit, equal to the line integral of A round the circuit, is given by

A1.2.11 $Φ = ℏ 2 e ∮ ∇ S ⋅ d r .$

Figure A1.2.2   Illustrating the quantization of flux.

The wavefunction ψ must be single-valued, and therefore, the line integral in Equation (A1.2.11) must be equal to 2πq, where q is an integer. Therefore, the trapped flux Φ can be nonzero, implying the existence of a magnetic field inside the cylinder, but the magnitude of the trapped flux is quantized in units of ϕ 0 = 2πћ/2e. The factor 2e here has its origin in the fact that the condensate is composed of electron pairs. This quantization of trapped flux is rather directly related to the existence of the condensate wavefunction.

If the external field is removed from the superconducting ring, the trapped flux remains. The superconductor is not then in a state of minimum possible free energy. It is, however, in metastable equilibrium, with an associated local minimum in the free energy. A transition to the state of absolutely minimum free energy would require that the flux passes through the superconductor and therefore penetrates it, at least transiently, by much more than the penetration depth, which is energetically very unfavourable (a consequence of the Meissner effect). More striking is the situation when the thickness, t, of the ring is much less than the penetration depth. A persistent current is still possible, although an extension of the argument in the preceding paragraph shows that the trapped flux is quantized in units less than 2πћ/2e. The metastability of this persistent current does not follow simply from the Meissner effect. It is necessary to note that a sudden loss of the trapped flux, or equivalently, loss of the persistent current, could occur only if all the Cooper pairs in the condensate were simultaneously to undergo a transition between two states (Equation [A1.2.9]) with different S( r ), which has negligible probability. Alternatively, the persistent current would disappear if it exceeds a critical value, equal to roughly Δ /p F, at which excitations are produced in such large numbers that superconductivity is suppressed (p F is the Fermi momentum); below this critical current, extra excitations may still be produced, the excitations being in a state of thermal equilibrium maintained by scattering, but the effect is only to reduce ns without destroying it altogether. Under certain circumstances, loss of a persistent current can occur with nonzero probability through the important process of gradual ‘phase slippage,’ which we shall discuss later. The persistence of the current then depends on phase slippage having a sufficiently small probability. We emphasize that the persistent current is an equilibrium state of the system (a minimum in the free energy, maintained by collisions), subject only to the constraint that the phase of C does not change.

#### A1.2.5  Flux Lines, the Mixed State, and Type I and Type II Superconductors

We might ask whether a single quantum of flux, for example, could be trapped inside a bulk sample of superconductor. It can be easily shown from Equation (A1.2.10) that in such a case the current would diverge to infinity along a line in the superconductor. In the presence of a very large current, it is energetically favourable for the Cooper pairs to break up, so the material would become effectively normal along this line. The situation can be described by an appropriate solution of the Ginsburg–Landau equations. The amplitude of the Ginsburg–Landau wavefunction vanishes along the line, and it rises to its normal value over a distance from the line of order the Ginsburg–Landau coherence length ξGL. The phase S( r ) changes by 2π as the line is encircled, so that one flux quantum is associated with the line. The magnetic field penetrates from the line to a distance of order of the penetration depth λ (Figure A1.2.3). The resulting structure is called a ‘flux line’ or ‘vortex,’ the latter name originating from the fact that close to the line the electron velocity is similar to that in a hydrodynamic vortex (proportional to 1/r). The ratio λ/ξGL is called κ. This ratio turns out to be approximately independent of temperature.

Figure A1.2.3   The modulus of the order parameter (| ψ |) and the magnetic flux density (B) plotted against radial distance (r) from the centre of a flux line (schematic).

Whether it is energetically favourable for such a structure to exist depends on its energy, ε, per unit length. Suppose that a superconductor in the form of a long thin solid cylinder (radius $≫ λ$ ) is exposed to a magnetic field, B, directed along its length. It can be shown that it is energetically favourable for a flux line to exist in the superconductor if B exceeds the value given by B = Bc 1 = ε/ϕ 0. We recall that if B exceeds the value given by Equation (A1.2.1) (the ‘thermodynamic critical field’) superconductivity is destroyed. Therefore, flux lines can be formed only if ε is sufficiently small, so that B c1 < B c. It turns out that this condition is equivalent to the condition that $κ > 1 2.$

If $κ > 1 2$ the superconducting cylinder passes directly from the ‘Meissner state’ (no field penetration except in the penetration depth) to the normal state when B exceeds B c. A superconductor of this type is called a type I superconductor; its phase diagram was shown in Figure A1.2.1. If $κ > 1 2$ flux lines can penetrate the superconducting cylinder at fields greater than B c1 (< B c), the diamagnetic moment of the cylinder being reduced. In fact more and more flux lines will penetrate until the (repulsive) interaction between them causes it to be no longer energetically favourable for them to form. As the applied field is increased, the density of flux lines increases, the diamagnetic moment falling, until the ‘cores’ of the flux lines (the regions of size ξLG where the superconductivity is suppressed) overlap, when the material becomes normal; the transition to the normal state occurs when $B > ∼ ϕ 0 / ξ LG 2$ which is equivalent to $B > B c 2 = 2 κ B c .$ A superconductor with $κ > 1 2$ is called a ‘type II superconductor.’ When a type II superconductor contains an array of flux lines, it is said to be in the mixed state. The magnetization curves for a type II superconductor is shown schematically in Figure A1.2.4, and the phase diagram for a type II superconductor is shown in Figure A1.2.5. (It is necessary to consider a long thin cylinder here because otherwise demagnetizing effects become important and result in a different type of behaviour. Even in type I superconductors, flux penetration can then occur through the formation of the intermediate state in which relatively large areas of normal state are embedded in the superconductor.)

Figure A1.2.4   Magnetization curve for a type II superconductor. Magnetization (M) plotted against applied magnetic flux density (B 0) (schematic).

Figure A1.2.5   Schematic phase diagram for a type II superconductor: applied magnetic flux density (B) plotted against temperature (T).

Type II superconductors are of great practical importance. Values of Bc are typically quite small (of order or less than 0.1 T), so that type I superconductors are useless in applications involving high magnetic fields. In contrast, values of Bc 2 can be very large, allowing, for example, the construction of superconducting coils for the generation of high magnetic fields. However, no pure single-element metal has a large value of κ, and indeed only niobium has a value of κ large enough (but only just) to make it type II. We are led, therefore, to look at other types of superconducting material, which we shall do in later sections.

#### A1.2.6  Phase Slip

The existence of flux lines allows us to see how the ‘persistent currents’ we described earlier can decay by ‘phase slippage.’ If a single flux line were to pass through the walls of the cylinder described in Section A1.2.4, the phase change round a circuit enclosing the hole in the cylinder would change by 2π. If the change is a decrease, the effect is a decrease in the persistent current. This phase slippage, which involves only a localized perturbation to ψ , can take place much more easily than any change involving simultaneously all the superconducting electrons. Creation of the flux line and its passage across the superconductor still generally encounters an energy barrier (some of it arising from an interaction between the flux line and the boundary of the superconductor). However, in the presence of a large enough current (the critical current), the barrier may be eliminated or reduced enough for thermally activated phase slippage to occur; the supercurrent can then decay in a relatively short time.

#### A1.2.7  Weak Links and Quantum Interferometers

This type of phase slippage often occurs in narrow constrictions in the superconductor, and we then talk of a weak link. A particularly simple type of weak link is the Josephson tunnel junction, formed by connecting two bulk volumes of superconductor through a thin insulating layer through which electrons can tunnel. The Cooper pairs can also tunnel, so that a supercurrent can pass across the junction. A simple analysis shows that for a weak junction the supercurrent is related to the difference in phase, ∆S, of the superconducting wavefunction across the junction by the relation

A1.2.12 $I = I 0 sin Δ S .$

The critical current is I 0, and we have described the dc Josephson effect. If I exceeds I 0, a potential difference, V, appears across the junction. A Cooper pair on one side of the junction then differs in energy by 2eV from one on the other side. Like an ordinary wavefunction, the superconducting wavefunction, ψ , has a time dependence exp(–iEt/ћ), so that the potential difference V gives rise to a continual slippage of the phase on one side of the junction relative to the other at a rate given by

A1.2.13 $d Δ S d t = 2 e V ℏ .$

The junction therefore carries an oscillating supercurrent of angular frequency 2eV/ћ (the ac Josephson effect). The phase slippage can be viewed as due to the steady flow of flux lines across the junction, each flux line causing a phase slip of 2π. The ac Josephson effect has been important in defining the volt in terms of frequency and the fundamental constants e and ћ.

Josephson junctions can be used to form quantum interferometers. Two junctions are arranged as shown in Figure A1.2.6 to form a superconducting loop. Simple quantum mechanics shows that, if a magnetic flux Φ threads the loop, the differences in phase across the two junctions must differ by 2πΦ/ϕ 0. The total critical current across the two junctions, I 0 (the maximum value of I 01sin∆S 1 + I 02sin∆S 2) must therefore be reduced by an amount that is periodic in Φ /ϕ 0. Since ϕ 0 is very small, we have the basis for a very sensitive magnetometer.

Figure A1.2.6   Schematic quantum interferometer.

#### A1.2.8  The Role of the Normal Electrons

We have seen that at a finite temperature there are condensed electrons in the superconductor and electronic excitations, which form, respectively, the superfluid and normal-fluid components in a two-fluid model. Any steady electric current in the superconductor is carried exclusively by the superconducting electrons; there is no electric field to drive the normal electrons, which exhibit an electrical resistivity due to the scattering of the excitations by lattice defects, as is the case for the electrons in an ordinary metal. At high frequencies, however, an electric field is required to move the superconducting electrons owing to their inertia, and this field will produce a response, and therefore dissipation, in the normal electrons. The superconductor is therefore not free from resistance at high frequencies, especially at microwave frequencies. In calculating this resistance, one must be careful to remember that the normal electrons are really excitations with different properties from electrons in the normal metal. At still higher frequencies (usually in the infrared), the photon energy may be sufficient to break a Cooper pair, leading to greatly increased dissipation.

The excitations play a role also in the thermal properties of the superconductor. Here, one must consider both the electronic excitations and the phonons. Both contribute to the heat capacity. Owing to the presence of the energy gap, the electronic contribution to the heat capacity becomes very small for $T ≪ T c$ , but the phonon contribution remains. The condensed electrons carry no entropy, so heat conduction is entirely due to the excitations. In normal pure metals, the heat is carried largely by the electrons; the phonons carry little heat because they travel much more slowly than the electrons and because they are strongly absorbed by the electrons. In a superconductor, the situation is very different, at least for $T ≪ T c$ . The excitations disappear at low temperatures, so they themselves carry little heat. At the same time the phonons can interact only with the electronic excitations (they do not have enough energy to break the Cooper pairs), so they are no longer strongly absorbed. Heat is therefore carried largely by the phonons, which can have a long mean free path as in a dielectric; the low-temperature phonon conductivity can therefore be high.

#### A1.2.9  Alloys

So far we have confined our attention to pure single-element metals, which are described by the BCS theory. The BCS theory is valid only if the electron–phonon interaction that is responsible for formation of the Cooper pairs is weak. In some metals, for example lead, this is not true, and a development of the BCS theory is required to treat such strong-coupling superconductors quantitatively. However, the basic physics remains unchanged.

For the rest of this chapter, we shall consider other types of superconducting material. As far as we know, superconductivity always involves the formation of electron pairs, but the pairs may be formed by a different (nonphonon) mechanism, and they may be formed in more complicated states than in a BCS superconductor, where the electrons in the pair have opposite spins and are in a state with no relative angular momentum (s-state).

We shall make a start, in this section, by considering a very important class of superconducting material formed from an alloy of two or more metals in which there is s-state pairing. In a pure metal, at the low temperatures required for conventional superconductivity, the electrons in the normal state have long mean free paths, much larger than the superconducting coherence length ξ0. In an alloy, this is generally no longer the case, owing to the scattering of electrons by the disorder in the alloy. At first sight, one might expect that this would have a profound effect, since it would surely affect the way in which the pair wavefunction can be formed. Surprisingly, it has very little effect in the case of nonmagnetic components. The addition of a nonmagnetic alloying element to a pure metal has very little effect on the critical temperature. It turns out that this is because the pairing wavefunction can be set up on the basis of the real normal-state electronic wave functions, whatever they are, but still with the pairing of opposite spins; in the case of an alloy, the wavefunctions include the effect of the scattering. The gap parameter as a function of temperature is essentially the same, as is the critical temperature. In many pure metals, the electronic wavefunctions in the normal state exhibit isotropy associated with the details of the band structure, and this is reflected in anisotropy in the gap parameter, this parameter varying round the Fermi surface in a way that is consistent with the crystal symmetry. Formation of the Cooper pairs from wavefunctions that describe an electron that is being continually scattered removes this anisotropy, but otherwise there is little effect on the gap parameter.

In more detail, however, the properties of an alloy do differ in important ways from those of a pure metal, especially if the electron mean free path in the normal state, , is less than ξ0 in the pure metal. The effective density of superconducting electrons is reduced by the scattering, by a factor of 0 in the limit $l ≪ ξ 0 ,$ so that the penetration depth is correspondingly increased. The coherence lengths are modified. The Pippard coherence length (governing the range of the nonlocal relation between J and A ) is reduced to in the same limit, while the Ginsburg–Landau coherence length at low temperatures becomes (ξ0)1/2. We see, therefore, that the value of κ is increased. Alloys tend, therefore, to be type II superconductors. Suitably chosen alloys can have large values of κ, with a resulting large value of the upper critical field Bc 2. Such alloys might therefore be used in applications in which the superconductor is exposed to a high magnetic field.

An important application is the production of high magnetic fields with a superconducting coil. However, a high κ is not sufficient for this purpose. In a high field, the superconducting material will be in the mixed state. When it carries an electric current, the flux lines will be subject to a force (roughly speaking a Lorenz force equal to ϕ 0 × J per unit length of line), which may cause the lines to move in the transverse direction, giving rise to phase slip and dissipation. To prevent such movement, the flux lines must be pinned by suitable microstructure in the alloy. Much of the technology of magnet materials is concerned with the enhancement of this pinning.

#### A1.2.10  Towards Higher Values of T c and H c2

Many practical applications would benefit from materials with higher values of both Tc and Hc 2. Among conventional superconductors, the best that can be done appears to be among intermetallic compounds with the cubic β-W (A15) structure and among the Chevrel phase compounds with composition M x Mo6X8, where M is a metal and X is either sulphur or selenium. Of these compounds, the best known is probably the A15 compound Nb3Sn, which has a critical temperature of about 18 K and an upper critical field at low temperatures exceeding 30 T, and which has been used extensively in superconducting magnets. These materials are of course strong-coupling superconductors, and the question arose whether stronger and stronger electron–phonon coupling can lead to higher and higher critical temperatures. Analysis that takes into account both the electron–phonon interaction and the Coulomb repulsion between electrons shows almost certainly that conventional mechanisms of superconductivity cannot lead to critical temperatures exceeding about 30 K, a view that seems to be confirmed by practical experience.

#### A1.2.11  Unconventional Superconductivity

So far, this introduction has been concerned with ‘conventional’ superconductors; i.e., those in which superconductivity arises from the formation of a condensate of electron pairs, the attractive electron–electron interaction required to produce the pairs being due to a distortion of the lattice (phonon exchange). In the simplest form of the BCS theory, the electron pairs form in states with zero internal angular momentum (s-states) and therefore with antiparallel spins, and conventional superconductors are correctly described by this form of the theory.

During the past 25 years or so, many superconducting materials have been discovered that appear not to be conventional. It appears that all involve electron pairing and, presumably, the formation of a condensate from the pairs, but they are unconventional in the sense that they involve pairing into states with nonzero angular momentum and/or a pairing mechanism that is not due to electron–phonon interaction.

A search for superconducting materials that do not depend on the electron–phonon interaction was based in part on the wish to find materials with higher critical temperatures. It was suggested by Little in 1964 that superconductivity might be found in solids composed of certain long-chain organic molecules, and that the electron–electron attraction might then arise through the electronic polarization of side-chain molecules. At that time even organic metals were hardly known, but over the years the search for, and study of, such materials has been intense, and it led eventually to the discovery of organic superconductors (although not in polymers). The normal metallic properties of these materials involve much interesting and novel physics (and chemistry!), connected, for example, with their low dimensionality, and the study and understanding of these normal properties have been important.

Much interest has been shown in the possibility that non-s-state pairing might be found in some superconductors. We recall that atoms of the light isotope of helium, 3He, are fermions and have a nuclear spin of one-half. The superfluid phase of liquid 3He was discovered in 1972 at temperatures less than a few mK, and it was shown very quickly that it could be described by a development of BCS theory that incorporated p-state pairing and parallel nuclear spins (L = 1, S = 1), and which had been developed, at least in part, in earlier years. Pairing in s-states is not possible because of the strong short-range repulsion between two helium atoms; p-state pairing makes use of the long-range attraction. Owing to the p-state pairing, which gives ‘structure’ to a Cooper pair, superfluid 3He is much more complicated than superfluid 4He, and it exhibits a rich variety of phenomena that have no counterpart in 4He. One aspect of this complication is the existence of (at least) two different superfluid phases of 3He; there is p-state pairing in both phases (A and B), but the orbital and spin angular momenta are differently arranged around the Fermi surface in the two cases. The nontrivial structure of a Cooper pair means that the order parameter has more components than in the case of s-state pairing: for p-state pairing, for example, the order parameter has in principle 18 components (three possible spin orientations; three possible orientations of the orbital angular momentum, and each of the possible nine combinations has its own real and imaginary components of the order parameter); Ginsburg–Landau theory becomes correspondingly more complicated. It should also be mentioned that the study of normal liquid 3He played an important role in the development of our understanding of normal Fermi liquids, a development that carried over into a much better understanding of the electrons in a normal metal than is possible in terms of a model in which the conduction electrons are regarded as a noninteracting Fermi gas. In the case of normal 3He, there are strong exchange interactions that cause the liquid to have a greatly enhanced Pauli susceptibility (due to the nuclear spins) and to be, in fact, almost a nuclear ferromagnet. As a result, there are, within the liquid, strong local fluctuations in the spin magnetic moment (paramagnons). Exchange of paramagnons contributes in an important way to the interatomic interaction leading to pairing and superfluidity, and it also serves to allow types of superfluid phase to exist that would not otherwise be stable (e.g., the A phase).

A superconductor with non-s-state pairing can exhibit properties differing from those of a conventional superconductor, which can allow us to identify it. In some phases of a non-s-state superconductor, the energy gap will vanish at certain points, or on certain lines, on the Fermi surface. This affects the temperature dependence of both the heat capacity and the normal fluid fraction (and hence the penetration depth); it gives rise to a power law dependence at the lowest temperatures instead of the exponential dependence that is characteristic of an energy gap which does not vanish in this way (it should be noted, however, that conventional superconductors containing magnetic impurities can also be gapless). However, this vanishing of the energy gap is not present in all phases of a non-s-state superfluid (e.g., the B-phase of superfluid 3He), and a true vanishing of the energy gap may be difficult to distinguish experimentally from a very strong anisotropy in an s-state system. As we have already noted, a non-s-state superconductor may exist in different superconducting phases, depending on the temperature and the applied magnetic field, as is the case in superfluid 3He; different phases are seen in some superconductors that are based on heavy fermion metals, suggesting non-s-state pairing in these cases (again the heavy fermion metals have interesting normal states, and similarities with liquid 3He are likely, spin fluctuations being important in both the normal and superconducting phases). A fairly clear-cut indication of unconventional pairing comes from the effect of impurity scattering. We noted earlier that in a conventional superconductor, such scattering has the effect of only smoothing out any anisotropy in the energy gap; in the case of unconventional pairing, the scattering mixes states with different arrangements of the spin and/or orbital angular momenta, which destroys the superconductivity, such destruction occurring when the mean free path associated with the scattering is less than the superconducting coherence length. Another fairly clear-cut indication of non-s-state pairing can come from tunneling studies. For example, d-state pairing (L = 2, S = 0) can lead to a situation where the order parameter has effectively a different sign on different parts of the Fermi surface; this can be observed in the behaviour of a quantum interferometer if the two junctions are formed with a second conventional superconductor on different sides of a single crystal of the unconventional superconductor. Finally, some types of unconventional pairing can involve a violation of time-reversal symmetry, so that the ground state can carry a nonzero (surface) current, even in the absence of an applied magnetic field.

#### A1.2.12  The High Tc Cuprates

The best known examples of what are almost certainly unconventional superconductors are the high T c cuprates: La2– x (Ca,Sr) x CuO4, with T c in the range 30–40 K, discovered by Bednorz and Müller in 1978; YBa2Cu3O7– x (YBCO) with T c = 93 K, discovered by Wu and co-workers in 1978; and all the others, C1, discovered over the subsequent years. Their potential applications have attracted enormous interest. But they are fascinating and important also in their basic physics. They differ from the classic conventional superconductor in at least four respects: their properties in the normal state are anomalous and not described by conventional Fermi liquid theory; they can have very high critical temperatures; they have layer structures, which lead to extreme anisotropy as between properties involving current flow within a layer and those involving current flow between layers; and they have very large (and anisotropic) values of κ, with very small (and anisotropic) coherence lengths, leading to inaccessibly high values of Bc 2. It seems unlikely that such high values of T c can result from a phonon-mediated attraction between electrons, but there is as yet no agreement about the mechanism. Indeed, there is no agreement about a theory of the normal state. The extreme anisotropy leads to interesting flux line structures, and this fact combined with the high T c allows observation of new effects in the mixed state, such as the melting of the flux line lattice and the decoupling of the parts of a vortex in different layers (formation of pancake vortices). Finally, some, at least of the cuprate superconductors, are almost certainly not s-state superconductors: tunneling experiments of the type described in the preceding section show that YBCO probably involves d-state pairing. The physics of these materials continues to pose major challenges, as does the development of the practical applications of them.

Many conventional and unconventional materials are discussed in this handbook, and the reader is encouraged to explore those here.

This version has been reviewed by Terry P. Orlando (MIT), with some minor suggested updates.

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