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In the previous sections we have looked at some of the key properties of materials in the superconducting state.  These include zero resistivity, restoration of the normal state under the action of a sufficiently high magnetic field, and the Meissner effect.  In this section and the next we will be concerned with trying to understand how and why the superconducting phase arises in the first place. 

Now, the conventional theory of conductors cannot be adapted to incorporate superconductivity.  According to the conventional theory, at low temperatures the impurities and defects in the lattice of ions will always give a non-zero contribution to the resistivity of a conductor, and yet we know that in the superconducting phase, the resistivity is zero. 

However, the properties of the lattice of ions in our superconducting material lie at the heart of the explanation of superconductivity.  There are various observations that hint at this.  These include: 

1.  Tin is a metallic element that can exist in different physical (allotropic) forms.  The so-called white tin is a superconductor, whereas grey tin (which has a different lattice structure) is not a superconductor. 

2.  Measurements show that the transition temperature T_C to the superconducting state varies for different isotopes of the same element.  (The mass of the lattice ions will vary between materials composed of different isotopes of the same element).  This variation of T_C with atomic mass is referred to as the “isotope effect”. 

In fact, the isotope effect provides strong evidence that it is the vibrations of the lattice ions that are important.  This is because the frequencies of the lattice vibrations vary with the mass of the lattice ions.  Experiments show that for an elemental superconductor, the transition temperature varies with the ion mass approximately according to 

where M is the mass of each ion in the lattice.  The natural frequency of vibration of an oscillating ion of mass M shows the same proportionality.  It can be shown that if the interactions between the electrons in the conductor and the lattice vibrations are important in determining the properties of the superconducting state, then the transition temperature should vary with ionic mass according to the relation above. 

So, the interaction between electrons and the lattice vibrations is a key factor in the onset and properties of the superconducting phase.  Now we need to ask how and why it is important. 

The point is that, in the superconducting phase, the lattice mediates an effective attractive interaction between pairs of electrons.  In the absence of the lattice, this effective interaction between electrons cannot exist.  Of course, the electrostatic interaction between electrons still exists in the presence of the lattice, but so – under the right conditions – does a much more subtle attractive interaction between the electrons. 

To understand how an effective interaction between electrons can arise in a lattice of ions, consider the following diagram: 

When an electron travels through the lattice, it exerts an electrostatic attraction on the positively charged ions in the lattice.  This causes the ions to be distorted away from their equilibrium positions in the lattice, towards the path of the electron (for example as in the middle section of the above the diagram).  This results in a region of net positive charge in the lattice. 

Now, the ions in the lattice are much more massive than the electrons, and as such they respond and change position more slowly than the electrons.  Thus, the distortion of the lattice remains for a while even when the initial electron is long gone from that part of the lattice. 

While the region of positive charge remains, a second electron can feel an electrostatic attraction towards that region (right-hand side of the above diagram).  In this way, the motion of the second electron is correlated with that of the first electron, resulting in an effective attractive interaction between the two electrons. 

The above description provides a very simplified physical picture of the nature of the effective interaction between electrons in a lattice, because we have not taken into account the effect of vibrations of the lattice.  A more sophisticated picture is shown in the figure below (adapted from figure 14.6 in Rosenberg’s book The Solid State, 1986). 

 In this model, an electron moving in the lattice (e.g. the upper electron in the above figure) might interact with one of the vibrating ions.  The effect of this interaction could be to modify the vibrations of the ion.  The effect of the modified vibrations of the ion could then be to attract a second electron (e.g. the bottom electron in the above figure) towards the ion. 

In the technical language of quantum mechanics and superconductivity, in this interaction the first electron is said to emit a virtual phonon, which is then absorbed by the second electron.  It can be shown that such an interaction conserves the momenta of the two electrons.  The result of this virtual phonon exchange is an effective attractive interaction between the two electrons.  The interaction is mediated by the vibrations of the ions, and cannot take place if the lattice of ions is not present. 

Herein lies the key to understanding superconductivity.  Pairs of electrons that interact as described above are called Cooper pairs (named after the physicist that first proposed their existence).  A proper description of Cooper pairs and their role in determining the properties of superconducting materials requires an advanced mathematical theory called the BCS (Bardeen Cooper Schrieffer) theory.  This theory is too complex to consider here, as it relies on advanced quantum mechanical and field-theoretical concepts.  However, in the next section, we will qualitatively discuss some of the properties of Cooper pairs and their role in the superconducting state.

Next:  Cooper Pairs