Superconductivity

"A theory of superconductivity is presented, based on the fact that the interaction between electrons resulting from virtual exchange of phonons is attractive when the energy difference between the electrons states involved is less than the phonon energy, ℏω. It is favorable to form a superconducting phase when this attractive interaction dominates the repulsive screened Coulomb interaction. The normal phase is described by the Bloch individual-particle model. The ground state of a superconductor, formed from a linear combination of normal state configurations in which electrons are virtually excited in pairs of opposite spin and momentum, is lower in energy than the normal state by amount proportional to an average (ℏω)2, consistent with the isotope effect. A mutually orthogonal set of excited states in one-to-one correspondence with those of the normal phase is obtained by specifying occupation of certain Bloch states and by using the rest to form a linear combination of virtual pair configurations. The theory yields a second-order phase transition and a Meissner effect in the form suggested by Pippard. Calculated values of specific heats and penetration depths and their temperature variation are in good agreement with experiment."

"Identifying the pairing mediators in the Fe-based superconductors is a necessary but far from sufficient step toward a full understanding of the materials’ superconductivity. Electrons in crystals are confined to bands shaped by the lattice’s constituent atoms and crystal structure. In the superconducting state, which is an intrinsically many-body state, the electrons must also obey Hund’s and other quantum rules. Those restrictions on electrons’ freedom of movement and association dictate perhaps the most eagerly sought characteristic of a new superconductor: its pairing symmetry."

"As you know, very many metals become superconducting below a certain temperature...—the temperature is different for different metals. When you reduce the temperature sufficiently the metals conduct electricity without any resistance. This phenomenon has been observed for a very large number of metals but not for all, and the theory of this phenomenon has caused a great deal of difficulty. It took a very long time to understand what was going on inside of superconductors, and I will only describe enough of it for our present purposes. It turns out that due to the interactions of the electrons with the vibrations of the atoms in the lattice, there is a small net effective attraction between the electrons. The result is that the electrons form together, if I may speak very qualitatively and crudely, bound pairs."

"The high-temperature copper-oxide superconductors, which offer resistance-free current flow at temperatures extending well above 100 K, are formed by doping certain copper-oxide compounds or by adding excess oxygen to them. The parent compounds—all antiferromagnetic insulators—couldn't be more different from their superconducting offspring: Magnetism and superconductivity are generally antithetical."

"I think that the single most important thing accomplished by the theory of John Bardeen, Leon Cooper, and Robert Schrieffer (BCS) was to show that superconductivity is not part of the reductionist frontier (Bardeen et al. 1957). Before BCS this was not so clear. For instance, in 1933 Walter Meissner raised the question of whether electric currents in superconductors are carried by the known charged particles, electrons and ions. The great thing that Bardeen, Cooper, and Schrieffer showed was that no new particles or forces had to be introduced to understand superconductivity. According to a book on superconductivity that Cooper showed me, many physicists were even disappointed that “superconductivity should, on the atomistic scale, be revealed as nothing more than a footling small interaction between electrons and lattice vibrations”. (Mendelssohn 1966)."

"The difference between Type I and Type II superconductivity is of substantial practical importance. To understand why, let us return to the Meissner effect ... What happens when the external magnetic field is increased to the critical value Bcrit? Magnetic flux begins to penetrate the material, initially in the form of flux lines. In the Type I case, the flux lines attract, forming a large region of normal metal, and superconductivity is soon lost altogether. In the case of a Type II superconductor, when one reaches the critical magnetic field, flux lines appear inside the superconductor. But since they repel each other, they can form a stable arrangement, a “lattice” of parallel flux lines ... As a result, a Type II superconductor for B > Bcrit can reach a stable arrangement in which part of an externally applied magnetic field is expelled to the outside world, while part penetrates the superconductor in the form of the flux lattice. The material remains superconducting in such a state, and as a result, Type II superconductors can support considerably higher magnetic fields and currents, making them more useful for many applications. Superconductivity is finally lost at a higher value of the magnetic field called the upper critical field."

"When certain materials are cooled below a certain critical temperature Tc, they suddenly become superconducting. Historically, physicists had long suspected that the superconducting transition, just like the superfluid transition, has something to do with Bose-Einstein condensation. But electrons are fermions, not bosons, and thus they first have to pair into bosons, which then condense. We now know that this general picture is substantially correct: Electrons form Cooper pairs, whose condensation is responsible for superconductivity. With brilliant insight, Landau and Ginzburg realized that without having to know the detailed mechanism driving the pairing of electrons into bosons, they could understand a great deal about superconductivity by studying the field φ(x) associated with these condensing bosons. In analogy with the ferromagnetic transition in which the magnetization M⃗(x) in a ferromagnet suddenly changes from zero to a nonzero value when the temperature drops below some critical temperature, they proposed that φ(x) becomes nonzero for temperatures below Tc. (In this chapter x denotes spatial coordinates only.) In statistical physics, quantities such as M⃗(x) and φ(x) that change through a phase transition are known as order parameters. The field φ(x) carries two units of electric charge and is therefore complex."