Previous:  Origin of Superconductivity

In the previous section we saw that, under certain conditions, electrons moving in an ionic lattice can experience an effective attractive force, such that electrons can be paired together to form the so-called Cooper pairs.  Of course, if the electrons are to remain paired, then they must undergo a continuous exchange of virtual phonons, mediated by the lattice. 

The BCS theory shows that Cooper pairs are held together with a certain “binding energy”, and can only undergo scattering in the ion lattice if the energy associated with the scattering process is sufficient to split up the Cooper pairs into their constituent electrons.  Otherwise, the Cooper pairs move through the lattice unscattered.  This is the mechanism by which the superconducting phase exhibits zero resistivity – if there is no scattering of Cooper pairs, then they do not experience electrical resistance in the superconducting material. 

One possible source of energy that could break the binding energy of a Cooper pair is the thermal vibrations of the ion lattice.  However, at low temperatures the thermal energy of the lattice is much lower than it is at higher temperatures.  Therefore, at low temperatures, the energy to break up the Cooper pairs is not available, and so the Cooper pairs remain intact.  This is why superconductivity is observed at very low temperatures. 

To put some flesh onto this, the fact that Cooper pairs are held together with a “binding energy” means that there is an energy gap between the Cooper pair states and the free electron states that the electrons would otherwise occupy in the normal conducting phase.  We can represent this schematically as in the following diagram: 

Thus, in order to break up a Cooper pair and return the two electrons to the free electron energy states, each electron must be supplied an energy Δ.  The BCS theory can be used to show that, at absolute zero, the energy gap in this figure is approximately 

where k is a physical constant known as Boltzmann’s constant.  It appears in theories that relate to the thermal properties of solids, liquids and gases, and it takes the value 

            k = 1.38 10^-23 J K^-1.  

For mercury, the critical temperature is about 4 K, and so the energy gap (at absolute zero) is approximately 

            2Δ ~ 3.5 × 1.38 10^-23 × 4 ~ 1.9 10^-22 joules = 0.001 eV. 

This is a fairly small amount of energy.  For example, the bound energy of an electron in the ground state of the hydrogen atom is 13.6 eV – over 10,000 times greater. 

The BCS theory can also predict how the energy gap 2Δ varies with temperature.  The result is a plot that looks something like that in the figure below. 

In this figure, 2Δ₀ is the energy gap at absolute zero, as we discussed above.  This plot is found to agree very well with the results of experimental measurements, for a wide range of different superconducting materials. 

One of the key features to note in this plot is that the energy gap becomes zero as the critical temperature is approached.  Thus, at the critical temperature and above it, the pairing of electrons into Cooper pairs is no longer energetically favourable, and the superconducting state is not observed at these temperatures. 

Finally, we note that the theory and properties of Cooper pairs extends well beyond what we have just discussed.  Unfortunately, to appreciate these aspects properly, it is necessary to have a good understanding of quantum mechanics and the other theoretical tools that are required to analyse the properties of Cooper pairs. 

Hopefully, however, the discussion in this section and the previous sections gives you a flavour of what superconductivity is all about and how the superconducting state arises.

THE END