The properties of Type I superconductors were modeled successfully by the efforts of John Bardeen, Leon Cooper, and Robert Schrieffer in what is commonly called the BCS theory. A key conceptual element in this theory is the pairing of electrons close to the Fermi level into Cooper pairs through interaction with the crystal lattice. This pairing results from a slight attraction between the electrons related to lattice vibrations; the coupling to the lattice is called a phonon interaction.
1) One of the first steps toward a theory of superconductivity was the realization that there must be a band gap separating the charge carriers from the state of normal conduction.
2) The critical temperature for superconductivity must be a measure of the band gap, since the material could lose superconductivity if thermal energy could get charge carriers across the gap.
3) The critical temperature was found to depend upon isotopic mass. It certainly would not if the conduction was by free electrons alone. This made it evident that the superconducting transition involved some kind of interaction with the crystal lattice.
5) The needed boson behavior was consistent with having coupled pairs of electrons with opposite spins. The isotope effect described above suggested that the coupling mechanism involved the crystal lattice, so this gave rise to the phonon model of coupling envisioned with Cooper pairs.
Pairs of electrons can behave very differently from single electrons which are fermions and must obey the Pauli exclusion principle. The pairs of electrons act more like bosons which can condense into the same energy level. The electron pairs have a slightly lower energy and leave an energy gap above them on the order of .001 eV which inhibits the kind of collision interactions which lead to ordinary resistivity. For temperatures such that the thermal energy is less than the band gap, the material exhibits zero resistivity.
Bardeen, Cooper, and Schrieffer received the Nobel Prize in 1972 for the development of the theory of superconductivity.
Bardeen, Cooper, and Schrieffer received the Nobel Prize in 1972 for the development of the theory of superconductivity.
Ideas Leading to the BCS Theory
The BCS theory of superconductivity has successfully described the measured properties of Type I superconductors. It envisions resistance-free conduction of coupled pairs of electrons called Cooper pairs. This theory is remarkable enough that it is interesting to look at the chain of ideas which led to it.
1) One of the first steps toward a theory of superconductivity was the realization that there must be a band gap separating the charge carriers from the state of normal conduction.
A band gap was implied by the very fact that the resistance is precisely zero. If charge carriers can move through a crystal lattice without interacting at all, it must be because their energies are quantized such that they do not have any available energy levels within reach of the energies of interaction with the lattice.
A band gap is suggested by specific heats of materials like vanadium. The fact that there is an exponentially increasing specific heat as the temperature approaches the critical temperature from below implies that thermal energy is being used to bridge some kind of gap in energy. As the temperature increases, there is an exponential increase in the number of particles which would have enough energy to cross the gap.
A band gap is suggested by specific heats of materials like vanadium. The fact that there is an exponentially increasing specific heat as the temperature approaches the critical temperature from below implies that thermal energy is being used to bridge some kind of gap in energy. As the temperature increases, there is an exponential increase in the number of particles which would have enough energy to cross the gap.
2) The critical temperature for superconductivity must be a measure of the band gap, since the material could lose superconductivity if thermal energy could get charge carriers across the gap.
3) The critical temperature was found to depend upon isotopic mass. It certainly would not if the conduction was by free electrons alone. This made it evident that the superconducting transition involved some kind of interaction with the crystal lattice.
4) Single electrons could be eliminated as the charge carriers in superconductivity since with a system of fermions you don't get energy gaps. All available levels up to the Fermi energy fill up.
5) The needed boson behavior was consistent with having coupled pairs of electrons with opposite spins. The isotope effect described above suggested that the coupling mechanism involved the crystal lattice, so this gave rise to the phonon model of coupling envisioned with Cooper pairs.
Experimental Support: BCS Theory
Electrons acting as pairs via lattice interaction? How did they come up with that idea for the BCS theory of superconductivity? The evidence for a small band gap at the Fermi level was a key piece in the puzzle. That evidence comes from the existence of a critical temperature, the existence of a critical magnetic field, and the exponential nature of the heat capacity variation in the Type I superconductors.
The evidence for interaction with the crystal lattice came first from the isotope effect on the critical temperature.
The band gap suggested a phase transition in which there was a kind of condensation, like a Bose-Einstein condensation, but electrons alone cannot condense into the same energy level (Pauli exclusion principle). Yet a drastic change in conductivity demanded a drastic change in electron behavior. Perhaps coupled pairs of electrons with antiparallel spins could act like bosons?
The evidence for interaction with the crystal lattice came first from the isotope effect on the critical temperature.
The band gap suggested a phase transition in which there was a kind of condensation, like a Bose-Einstein condensation, but electrons alone cannot condense into the same energy level (Pauli exclusion principle). Yet a drastic change in conductivity demanded a drastic change in electron behavior. Perhaps coupled pairs of electrons with antiparallel spins could act like bosons?
Measured Superconductor Bandgap
The measured bandgap in Type I superconductors is one of the pieces of experimental evidence which supports the BCS theory. The BCS theory predicts a bandgap of
where Tc is the critical temperature for the superconductor. The energy gap is related to the coherence length for the superconductor, one of the two characteristic lengths associated with superconductivity.
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