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One of the best known laws in physics is Ohm’s Law, which relates the flow of current through a material to the potential difference across the material.  To illustrate this, consider the following figure, which shows the effect of applying an electric field to a section of a good conductor, for example a piece of copper. 

A conducting material such as copper consists of a lattice of ions that sit in a “gas” of electrons, the electron gas being composed of the outer electrons of the copper atoms that make up the solid.  The spacing of the copper ions is about 2.5 10^-10 m.  When the electric field is applied, the electrons experience a force that acts in the opposite direction of the applied field.  Thus, there is a net drift of electrons to the right in the figure above, and an electric current flows in the conductor. 

Ohm’s law states that the current density J (current per unit area of conductor in the direction of current flow) is related to the electric field E through the relation 

where σ is known as the conductivity of the material.  Now, if the electric field is constant, then the potential difference across the conductor is given V = EL.  The total current flow in the conductor is equal to I = JA.  Substituting into the expression above gives a more familiar form of Ohm’s Law:

 where the “resistance” R is given by 

In this expression, ρ is called the “resistivity”, and is the quantity that we will be interested in from now on.  Note that the resistivity depends only on the nature of the conducting material, whereas the resistance is also depends on the dimensions of the conductor. 

Now, when the electric field is applied, the electrons in the conductor undergo an acceleration against the direction of the field.  But since the current in the conductor remains finite and rapidly attains a constant value (given by Ohm’s law), the electrons must be subject to processes within the conductor that oppose the acceleration they experience under the influence of the electric field.  That is, when the current reaches its constant value, the acceleration of the electrons is balanced by retarding mechanisms in the conductor.  The magnitude of these retarding mechanisms is reflected in the value of the electrical resistivity. 

There are three candidate retarding mechanisms: 

1.  Interaction of the electrons with the positive ions in the lattice; 

2.  Interaction of the electrons with impurities and defects in the conducting material; 

3.  Interaction of the electrons with the thermal vibrations of the ions in the lattice. 

In this context, impurities and defects refer to the presence (or absence) of other types of ion in the lattice that interfere with the symmetry of the lattice of ions.  The electron interaction mechanisms that we are considering here are usually referred to as electron “scattering” mechanisms, since one of the effects of the interactions is to adjust the trajectories of the electrons in the conductor. 

It turns out that only the second and third of the above mechanisms make a significant contribution to the electrical resistivity.  Scattering from the ions themselves makes a negligible contribution to electrical resistivity.  It is possible to do a calculation to estimate how frequently electrons are scattered by direct interactions with ions in the lattice.  Such a calculation shows that electrons scatter from about 1 in every 100 ions that they encounter while moving through the conductor.  Thus, the likelihood of an interaction with an ion is very low. 

The reason for this is related to the symmetry of the lattice of ions – in a perfect lattice the electrons travel unscattered because the electric forces felt by the electrons is the same in all directions.  Consideration of the wave properties of the electrons shows that in this situation, the electrons move unscattered. 

The second and third mechanisms contribute to the electrical resistivity because they break the symmetry of the lattice of ions.  Of particular interest is the interaction of the electrons with the thermal vibrations of the lattice of ions.  When the ions vibrate, for much of the time they are not located at their stationary positions.  This is shown in the following figure, which shows a snapshot of the location of the ions in the lattice at some given time. 

In this situation, the symmetry of the lattice is broken, and it is this loss of symmetry that provides the basis for an electron scattering mechanism.  When we come to look at the mechanism by which the superconducting state comes about, we will return to the question of how electrons can interact with lattice vibrations. 

The electrical resistivity of a conductor such as copper, as a function of temperature, is similar to that shown in the following diagram. 

Since the arrangement of the impurities and defects in the lattice do not change with temperature, their contribution to electrical resistivity is independent of temperature.  However, as temperature increases, the amplitude of the lattice vibrations increases.  The contribution to resistivity due to electron interactions with lattice vibrations therefore increases with temperature.  It can be shown that at high temperatures, the contribution to resistivity is proportional to the temperature.  At lower temperatures, the contribution is proportional to the fifth power of temperature. 

In our survey of superconductivity, it is this dependence of electrical resistivity on temperature that is of interest, in particular how the electrical resistivity behaves as the temperature approaches absolute zero.  For non-superconducting materials, the form of the electrical resistivity as a function of temperature is as shown above.

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