Superconductivity is a fascinating phenomenon, which exhibits the effects of quantum mechanics on a macroscopic scale. In particular, the electrons move coherently in a single quantum state. In conventional superconductors, this coherence has come about because of an effective attraction between electrons due to the lattice vibrations (phonons) and sees the electrons paired together (as Cooper pairs). Cooper pairs have integer spin, so have the properties of bosons, and indeed, superconductivity can be considered as the Bose-Einstein condensation of Cooper pairs.
Superconductors are named as such, because of their ability to conduct electricity with zero energy loss, or zero resistance (up to a critical current density). This is principally because the current carrying state is also a quantum ground state of the superconductor, and there is a finite energy gap (Delta) to any excited state. This means that individual electrons are scattered by impurities or thermal vibrations, which are the normal causes of resistance. At temperatures high enough such that the thermal energy (kT) is of the order of the gap (Delta) the superconducting properties disappear. Hence superconductors only work below a critical temperature, Tc, which ranges from 0-23 degrees Kelvin for conventional superconductors, but has reached over 130K for high-temperature superconductors. Note that high-temperature in this field is still at colder than -140 degrees Celsius or -220 degrees Fahrenheit!
The more fundamental signature of a superconductor, though, is its ability to expel a magnetic field from within its interior, so long as the field is below a critical value (Hc). The expulsion of magnetic field is called the Meissner Effect, and results in a repulsive force between a superconductor and any magnet placed near it. Many demonstrations of superconductors show the Meissner Effect as a small magnet levitating above the superconductor. The effect has a large scale technological application in Japan, where it is used to levitate high speed trains.
De Haas - van Alphen Oscillations in the Vortex State.
Many superconductors are of type-II and can exist over a wide range of magnetic fields, between a lower critical field (Hc-1) where magnetic flux first enters the superconductor and an upper critical field (Hc-2) where the superconductivity is finally destroyed. In this region is the vortex state, as the magnetic field enters the superconductor in a regular array of flux vortices (it is also known as the Abrikosov lattice from the Russian who first predicted the state).
De Haas - van Alphen oscillations have been used for many years to study the Fermi surface of electrons. They are the result of the quantization of energy levels of electrons which orbit in a magnetic field. The level spacing is proportional to the magnetic field, and the number of electrons that can be in any one level increases as the orbits get smaller, which also corresponds to an increasing magnetic field. So at small magnetic fields, the electrons are in a large number of closely spaced levels, while in large fields they are in a small number of widely spaced levels. Between these extremes, as the field is increased, the levels all move up in energy, while the uppermost filled level loses its electrons. The energy of the system oscillates during the process, the oscillations being with regular period in the inverse magnetic field, and proportional to the area of the Fermi surface, which gives the details of the orbits of the highest energy electrons.
Observations have been made of de Haas - van Alphen Oscillations in the Vortex State, but with damped amplitude. A superconducting gap at the Fermi surface has been shown to reduce the amplitude of oscillations, and the existence of the vortex lattice is known to disturb the orbits of electrons (Landau orbits) which also reduces the amplitude of oscillations. Much theoretical work is in progress in an attempt to derive information about the details of the superconducting gap from the observed damping.
Strong Electron-Phonon Coupling Theory.
The original BCS theory of superconductivity, while giving a correct qualitative understanding of the mechanisms leading to superconductivity, contains a number of gross approximations which lead to incorrect estimates of quantities such as critical field and critical temperature, for a given value of the energy gap. The strong coupling theory of Migdal and Eliashberg goes beyond this, by including the phonon interactions in a more detailed manner, such that the electronic Green's functions (which indicate how the electrons move around) are renormalized (meaning that quantities such as the effective mass are changed - electrons "feel heavier" so are slowed down by interactions).
The EPR paradox appears to show that certain "common sense" questions about an electron (or other "quantum" particle) have no meaning until an observation is made. Such a positivist interpretation is unsatisfactory, but generally ignored. The Dirac equation, which is the fundamental description of electron, as it is relativistic and produces the spin properties of electrons, contains an infinity of solutions which are of negative energy, and assumed to be occupied by an infinite number (or "Dirac sea") of electrons, which do not interact, but whose absence is observed as a "hole" or positron. This state of affairs may be considered unsatisfactory by Occham's rasor, if a simpler solution is at hand. Both problems may be resolved if a more time-symmetric theory is established, and boundary conditions in time are considered in an equivalent manner to boundary conditions in space, as Einstein's special relativity might suggest. See John Gribbin's interesting discussions.